On the role of Convexity in Isoperimetry, Spectral-Gap and Concentration

Introduction

The first way is by means of an isoperimetric inequality. Recall that Minkowski’s (exterior) boundary measure of a Borel set $A\subset\Omega$, which we denote here by $\mu^{+}(A)$, is defined as:

A very useful isoperimetric inequality was considered by Cheeger (and in a more general form, independently by V. G. Maz’ya ):

The space $(\Omega,d,\mu)$ is said to satisfy Cheeger’s isoperimetric inequality if:

The best possible constant $D$ above is denoted by $D_{Che}=D_{Che}(\Omega,d,\mu)$.

A second way to measure the interplay between $d$ and $\mu$ is given by functional inequalities. Let $\mathcal{F}=\mathcal{F}(\Omega,d)$ denote the space of functions which are Lipschitz on every ball in $(\Omega,d)$ - we will call such functions “Lipschitz-on-balls” - and let $f\in\mathcal{F}$. We will consider functional inequalities which measure the relation between $\left\|f\right\|_{L_{p}(\mu)}$ and $\left\|\left|\nabla f\right|\right\|_{L_{q}(\mu)}$, for $0<p,q\leq\infty$ (more general Orlicz norms will be treated in ). Here, the effect of the metric $d$ is via the Riemannian metric $g$ which is used to measure $\left|\nabla f\right|:=g(\nabla f,\nabla f)^{1/2}$, although more general ways exist to define $\left|\nabla f\right|$ in the non manifold setting. Of course if $f$ is constant there is no sense to compare against $\left\|\left|\nabla f\right|\right\|_{L_{q}(\mu)}=0$, so we will need to exclude these cases. To this end, we will require that either the expectation $E_{\mu}f$ or median $M_{\mu}f$ of $f$ are 0. Here $E_{\mu}f=\int fd\mu$ and $M_{\mu}f$ is a value so that $\mu(f\geq M_{\mu}f)\geq 1/2$ and $\mu(f\leq M_{\mu}f)\geq 1/2$.

A well known example of a functional inequality was studied by Poincaré:

The space $(\Omega,d,\mu)$ is said to satisfy Poincaré’s inequality if:

The best possible constant $D$ above is denoted by $D_{Poin}=D_{Poin}(\Omega,d,\mu)$.

It is well known (e.g. ) that under appropriate smoothness assumptions, Poincaré’s inequality is equivalent to the existence of a spectral gap of an appropriate Laplacian operator $-\Delta_{g,\mu}$ on $(M,g)$ associated to the measure $\mu$ with corresponding boundary conditions on its support. When $\mu$ is uniform on a domain $\Omega\subset(M,g)$, $\Delta_{g,\mu}$ coincides with the usual Laplace-Beltrami operator $\Delta_{g}$ with Neumann boundary conditions on $\Omega$. The first non-trivial eigenvalue of $-\Delta_{g,\mu}$ (the “spectral gap”) is then precisely $D_{Poin}^{2}(\Omega,d,\mu)$.

A third way to measure the relation between $d$ and $\mu$ is given by concentration inequalities. These measure how tightly $1$-Lipschitz functions are concentrated about their mean, by providing a quantitative estimate on the tail decay $\mu(|f-E_{\mu}f|\geq t)$. A typical situation is given by the following example:

The space $(\Omega,d,\mu)$ is said to have exponential concentration if:

Fixing $c=e$, the best possible constant $D$ above is denoted by $D_{Exp}=D_{Exp}(\Omega,d,\mu)$. The best constant for a specific $f$ is denoted by $D_{Exp}(f)$.

It is known that the three examples mentioned above are arranged in a hierarchy. It was shown by Cheeger , and in a more general form, independently by Maz’ya (see also ), that Cheeger’s isoperimetric inequality always implies Poincaré’s inequality (or spectral gap):

$D_{Poin}\geq D_{Che}/2$ (“Cheeger’s inequality”).

The fact that Poincaré’s inequality implies exponential concentration was first shown by M. Gromov and V. Milman in the Riemannian setting, and subsequently by other authors in other settings as well (e.g. , see and the references therein):

There exists a universal numeric constant $c>0$ such that $D_{Exp}\geq cD_{Poin}$.

2 Reversing the Hierarchy

In both cases, we will say that “our convexity assumptions are fulfilled”. More generally, we present the following definition:

We will say that our smooth convexity assumptions are fulfilled if:

$d$ denotes the induced geodesic distance on $(M,g)$.

$d\mu=\exp(-\psi)dvol_{M}$, $\psi\in C^{2}(M)$, and as tensor fields on $M$:

We will say that our convexity assumptions are fulfilled if $\mu$ can be approximated in total-variation by measures $\left\{\mu_{m}\right\}$ so that $(\Omega,d,\mu_{m})$ satisfy our smooth convexity assumptions.

The condition (1.1) is the well-known Curvature-Dimension condition $CD(0,\infty)$, introduced by Bakry and Émery in their influential paper (in the more abstract framework of diffusion generators). Here $Ric_{g}$ denotes the Ricci curvature tensor and $Hess_{g}$ denotes the second covariant derivative. When the Ricci tensor satisfies a slightly relaxed condition $Ric_{g}\geq-Kg$, $K\geq 0$, it was first shown by Buser that the implication in Theorem 1.1 can be reversed. We only quote the $K=0$ case, which in our setting reads:

If $\mu$ is uniform on a closed $n$-dimensional manifold $(M,g)$ and $Ric_{g}\geq 0$ then $D_{Che}\geq cD_{Poin}$, where $c>0$ is a universal numeric constant.

The fact that the constant $c$ above does not depend on the dimension $n$ is quite remarkable. Buser’s theorem was recently further generalized by M. Ledoux (following the method developed by Bakry–Ledoux ) to the Bakry-Émery abstract setting. Again, we only quote the $CD(0,\infty)$ case:

Under our smooth convexity assumptions $D_{Che}\geq cD_{Poin}$, where $c>0$ is a universal numeric constant.

3 Main Theorem

How about reversing the implication in Theorem 1.2 under our convexity assumptions? This is one of the statements in our Main Theorem below. A second statement, which is much more surprising, concerns a very weak type of concentration inequality, which we introduce:

The space $(\Omega,d,\mu)$ is said to satisfy First-Moment concentration if:

The best possible constant $D$ above is denoted by $D_{FM}=D_{FM}(\Omega,d,\mu)$.

Clearly, by the Markov-Chebyshev inequality, First-Moment concentration implies linear tail-decay:

and decay slightly faster than linear implies (integrating by parts) First-Moment concentration. The First-Moment concentration is clearly a-priori much weaker than exponential concentration. Our Main Theorem, first announced in , asserts that under our convexity assumptions, not only is First-Moment concentration equivalent to exponential concentration, but in fact also to the a-priori stronger inequalities of Poincaré and Cheeger:

Under our convexity assumptions, the following statements are equivalent:

Cheeger’s isoperimetric inequality (with $D_{Che}$).

Exponential concentration inequality (with $D_{Exp}$).

First Moment concentration inequality (with $D_{FM}$).

The equivalence is in the sense that the constants above satisfy $D_{Che}\simeq D_{Poin}\simeq D_{Exp}\simeq D_{FM}$.

Here and below, $A\simeq B$ means that $C_{1}B\leq A\leq C_{2}B$, with $C_{i}>0$ some universal numerical constants, independent of any other parameter, and in particular the dimension $n$. We will see in Section 4 that the use of the First-Moment is not essential in Statement (4); we may have required any arbitrarily slow uniform tail decay, instead of linear decay. In other words, if:

where $\alpha$ decays to 0 arbitrarily slow, we can deduce under our convexity assumptions that Lipschitz functions have in fact much faster exponential tail decay (with rate depending solely on $\alpha$), and in addition the stronger inequalities of Poincaré and Cheeger, as above. In this sense, our result extends the well-known Kahane-Khinchine type inequalities in Convexity Theory (e.g. consequences of Borell’s Lemma , see for an overview) stating that linear functionals have comparable moments, ensuring exponential tail decay, to the same statement for the “worst” $1$-Lipschitz function (see Remark 4.4).

The Main Theorem may also be interpreted as stating that under our convexity assumptions, there exists a single $1$-Lipschitz function $f$ whose level sets on average attain the minimum (up to constants) in Cheeger’s isoperimetric inequality (see Section 4). In fact, one may choose this function to be of the form $f(x)=d(x,A)$, where $A$ is some set with $\mu(A)\geq 1/2$. This is expressed in the following reformulation of the Main Theorem:

Under our convexity assumptions on $(\Omega,d,\mu)$:

Equivalently, this is tantamount to saying that under our convexity assumptions, it is only necessary to use test functions of the form $f(x)=d(x,A)$ when testing (up to a universal numeric constant) for the spectral gap $D_{Poin}^{2}$ in Poincaré’s inequality. Clearly, without any further assumptions, all of the above statements are in general false.

4 Applications to Spectral Gap of Convex Domains

where $c>0$ is some universal numeric constant.

Here Vol denotes the Lebesgue measure. In particular, we see that:

Note that $K,L$ satisfying the above condition can be very different geometrically (consider for instance a Euclidean ball of radius $1$ and its intersection with a centered slab of width $10/\sqrt{n}$), and yet share essentially the same spectral gap. Also note that our stability result holds with respect to all possible Euclidean structures $|\cdot|$ simultaneously, since the assumption in the left-hand side above is independent of the Euclidean structure.

We also observe that the quantitative dependence on $v_{K},v_{L}$ in (1.4) is essentially best possible: the logarithmic dependence on $1/v_{L}$ is (up to numeric constants) optimal, and the quadratic dependence on $v_{K}$ cannot be improved beyond linear (and is in fact optimal in some restricted range, see Example 5.6). In addition, Theorem 1.7 implies that when $\frac{1}{a}L\subset K\subset Lb$ with $a,b\geq 1$, $ab\leq 1+\frac{c}{n}$, then $D_{Che}(K)\simeq D_{Che}(L)$. In fact, when $ab\leq 1+\frac{s}{n}$ with $1\leq s\leq n$, we obtain in Corollary 5.3 the best possible (up to numeric constants) quantitative bounds on $D_{Che}(K)/D_{Che}(L)$ as a function of $s$ (see Example 5.7). To the best of our knowledge, no quantitative bounds on the stability of $D_{Che}$ for convex domains under convex perturbations were previously known. Completely analogous stability results hold for log-concave probability measures as well (see Theorem 5.5). Another useful result which we deduce from our Main Theorem is that Cheeger’s constant is preserved under maps which are not necessarily Lipschitz, but rather Lipschitz on average (see Theorem 5.9).

5 Ingredients in Proof of Main Theorem

All of the four statements in our Main Theorem 1.5 can be equivalently (up to universal constants) rewritten using a single unified framework in terms of $(p,q)$ Poincaré inequalities:

The space $(\Omega,d,\mu)$ is said to satisfy a $(p,q)$ Poincaré inequality if:

The best possible constant $D$ above is denoted by $D_{p,q}=D_{p,q}(\Omega,d,\mu)$.

We prefer to use the median $M_{\mu}$ in our definition for reasons which will become apparent in Section 2. It is known and easy to establish that $D_{Poin}\simeq D_{2,2}$, $D_{Che}=D_{1,1}$, $D_{FM}\simeq D_{1,\infty}$, so our Main Theorem can be restated as the claim that all $(p,q)$ Poincaré inequalities in the range $1\leq p\leq q\leq\infty$ are equivalent under our convexity assumptions (see Theorem 2.4).

The convexity assumptions are used in an essential way in the proof of the Main Theorem in several separate places. First, we employ the $CD(0,\infty)$ condition via the semi-group gradient estimates used by Ledoux in his proof of Theorem 1.4. Contrary to previous approaches, which could only deduce isoperimetric information from functional inequalities with a $\left\|\left|\nabla f\right|\right\|_{L_{q}(\mu)}$ term with $q=2$ (see [8, p. 3] and the references therein), we can handle arbitrary $q\geq 1$ (and although we do not pursue this direction here, more general Orlicz norms too). To demonstrate that our estimates are sharp, we remark that the isoperimetric inequalities we obtain are in fact equivalent (up to universal constants) to the $(p,q)$ Poincaré inequalities used to derive them. This is summarized in Theorem 2.9, which generalizes Theorems 1.1, 1.2, 1.3 and 1.4 above into a single unified framework. Using this, we deduce from the First-Moment inequality ($p=1,q=\infty$ above) that:

To deduce Cheeger’s isoperimetric inequality from (1.5), we need to use our convexity assumptions for the second time. We employ the following series of results in Riemannian Geometry, due to numerous groups of authors , who proved them under increasingly general conditions. A detailed survey of these results may be found in the Appendix. We learned about these results from the PhD Thesis of V. Bayle , which was referenced to us by Sasha Sodin, to whom we are indebted. In the formulation below, we use a slightly more general notion of smooth convexity assumptions, which is defined in Section 6.

Under our generalized smooth convexity assumptions, the isoperimetric profile $I=I_{(\Omega,d,\mu)}$ is concave on $(0,1)$. Moreover, when $\mu$ is in addition uniform on $\Omega\subset(M,g)$, then $I^{n/(n-1)}$ is concave on $$, where$n$is the dimension of$M$.

It is not hard to show (see Section 6) that the isoperimetric profile $I$ is continuous under very general assumptions. It then follows by a general argument (e.g. Corollary 6.5) that $I$ must be symmetric about the point $1/2$. Hence, the concavity of $I$ implies that $D_{Che}=2I(1/2)$ under our convexity assumptions. It is then immediate to deduce Cheeger’s isoperimetric inequality from (1.5). In fact, a stronger statement can be deduced when $\mu$ is uniform on $\Omega$ (see Remark 2.11).

A final ingredient in the proof is an approximation argument to handle non-smooth densities, which are typical in applications as well as essential for handling uniform measures on bounded domains (with possibly non-smooth boundaries). Contrary to many results in Convexity Theory, where approximation arguments are standard, easy and usually omitted, the isoperimetric profile and the Cheeger constant are delicate objects, which in general are not stable under approximation in the natural total-variation metric (see Section 6). We therefore employ our convexity assumptions one last time, and provide in Section 6 a careful argument for deducing the Main Theorem 1.5 without any smoothness assumptions, and a different approximation procedure for extending Theorem 1.8, which in particular applies to the entire class of log-concave measures in Euclidean space.

The rest of this work is organized as follows. In Section 2, we reformulate the Main Theorem in terms of an equivalence between $(p,q)$ Poincaré inequalities, and using Theorem 1.8, reduce it to the statement of Theorem 2.9. The semi-group argument for proving Theorem 2.9 is described in Section 3. Further interpretations and an extension of the Main Theorem are described in Section 4. Applications for the spectral gap under our convexity assumptions are described in Section 5. We conclude with an approximation argument for disposing of our smoothness assumptions in Section 6, and an Appendix describing in more detail the results summarized in the statement of Theorem 1.8.

Acknowledgements. I would like to thank Professor Gideon Schechtman and the Weizmann Institute of Science where this research project commenced during the last months of my PhD studies. I would also like to thank Professor Jean Bourgain and the Institute for Advanced Study for providing the perfect research environment. Most especially, I would like to thank Sasha Sodin for his invaluable help - acquainting me with capacities, suggesting to look at the PhD Thesis of Bayle and Ledoux’s semi-group argument, countless other references, many informative conversations and comments on this manuscript. I am also grateful to Franck Barthe for his kind hospitality, remarks and advice, Bo’az Klartag for several references, discussions and insightful remarks, Professors Sergey Bobkov and Michel Ledoux for their comments and advice, and Professor David Jerison for several interesting conversations and suggestions. My gratitude also extends to the anonymous referees for their extraordinary careful reading and helpful suggestions, which greatly improved the presentation of this work. Finally, I would also like to thank Professors David Jerison, Erwin Lutwak, Assaf Naor, Vladimir Pestov and Santosh Vempala for their invitations to give talks on this work in its early development.

(p,q)𝑝𝑞(p,q) Poincaré Inequalities

We start by rewriting some of the statements of the Main Theorem 1.5.

Let $N(\mu)$ denote an Orlicz norm associated to the Young function $N$. Then:

Note that $\left\|1\right\|_{N(\mu)}=1/N^{-1}(1)$. First, by Jensen’s inequality (applied twice):

Next, we may assume that $M_{\mu}f\geq E_{\mu}f$ (otherwise exchange $f$ by $-f$). By the Markov-Chebyshev inequality:

We conclude by noting that $\frac{N^{-1}(2)}{N^{-1}(1)}\leq 2$ since $N$ is convex. ∎

The last lemma implies that we can pass back and forth between using the median $M_{\mu}$ and the expectation $E_{\mu}$ when excluding constant functions in our functional inequalities, at the expense of losing a universal constant. We therefore see that Poincaré’s inequality is equivalent (up to constants) to the inequality:

(and in fact in this case one clearly has $D_{Poin}\geq D^{M}_{Poin}$). The next lemma, due to Maz’ya and Federer and Fleming (see also for a careful derivation), rewrites Cheeger’s isoperimetric inequality in functional form:

Cheeger’s isoperimetric inequality (with $D_{Che}$) holds iff:

It is easy to show that Cheeger’s isoperimetric inequality is recovered by applying (2.2) to Lipschitz functions which approximate $\chi_{A}$, the characteristic function of a Borel set $A$, in an appropriate sense. Conversely, the co-area formula, which for general metric probability spaces becomes an inequality (see ), implies for $f\in\mathcal{F}$ with $M_{\mu}f=0$:

Since for a 1-Lipschitz function $f$, $\left\|\left|\nabla f\right|\right\|_{L_{\infty}(\mu)}\leq 1$, our First-Moment inequality is clearly equivalent to:

in the sense that $D_{FM}\simeq D^{M}_{FM}$ where $D^{M}_{FM}$ is the best constant above.

The above functional reformulations remain valid for general metric probability spaces $(\Omega,d,\mu)$, in which case we interpret $\left|\nabla f\right|$ for any $f\in\mathcal{F}$ as the following Borel function:

(and we define it as 0 if $x$ is an isolated point - see [19, pp. 184,189] for more details).

With the above reformulations (2.1), (2.2), (2.3) serving as motivation, the reasons behind our definition of $(p,q)$ Poincaré inequalities in the Introduction are now clear. Note that $D_{Che}=D_{1,1}$, $D^{M}_{Poin}=D_{2,2}$ and $D^{M}_{FM}=D_{1,\infty}$. We can now restate our Main Theorem 1.5 as follows:

Under our convexity assumptions, all $(p,q)$ Poincaré inequalities are equivalent in the range $1\leq p\leq q\leq\infty$. More precisely, for any other $1\leq p^{\prime}\leq q^{\prime}\leq\infty$:

In fact, a more precise dependence on $p$ and $p^{\prime}$ may be obtained in some cases. For instance, clearly $D_{p^{\prime},q^{\prime}}\geq D_{p,q}$ if $p^{\prime}\leq p$ and $q^{\prime}\geq q$ without any further convexity assumptions (by Jensen’s inequality), so we see that the First-Moment inequality ($(1,\infty)$ case) is the weakest among all $(p,q)$ Poincaré inequalities in the above range. Another immediate observation is given by:

Let $0<p\leq p^{\prime}\leq\infty$ and $0<q\leq q^{\prime}\leq\infty$ be such that:

Then without any further convexity assumptions, $D_{p^{\prime},q^{\prime}}\geq\frac{p}{p^{\prime}}D_{p,q}$.

Let $g\in\mathcal{F}$ denote a function with $M_{\mu}g=0$. Define $f=\text{sign}(g)|g|^{p^{\prime}/p}$, and apply the $(p,q)$ Poincaré inequality to $f$. Clearly $M_{\mu}f=0$, so we obtain by Hölder’s inequality:

Maz’ya–Cheeger inequality: $D_{Poin}\geq D_{Che}/2$.

Gromov–Milman inequality: $D_{Exp}\geq cD_{Poin}$.

Since $D_{Poin}\simeq D_{2,2}$, we conclude by Proposition 2.5 that $D_{p,p}\geq cD_{Poin}/p$ for every $2\leq p\leq\infty$. Let $f$ be a 1-Lipschitz function. It is elementary to show (e.g. ) that $1/D_{Exp}(f)$ is equivalent (to within universal constants) to $\left\|f-E_{\mu}f\right\|_{\Psi_{1}(\mu)}$, and that $\left\|g\right\|_{\Psi_{1}(\mu)}$ is in turn equivalent to $\sup_{p\geq 1}\left\|g\right\|_{L_{p}(\mu)}/{p}$. Employing Lemma 2.1 and using the $(p,p)$ Poincaré inequalities:

since $f$ was assumed 1-Lipschitz. Taking supremum on all such functions $f$, we obtain the conclusion. ∎

The exact same proof shows that $D_{Exp}\geq c_{r}D_{r,r}$, for arbitrary $r\geq 1$.

We have seen that passing from $(p,q)$ to $(p^{\prime},q^{\prime})$ is manageable if $q^{\prime}\geq q$ (perhaps under some additional assumptions on $p,p^{\prime}$) without any convexity assumptions. Unfortunately, we are interested in the case $q^{\prime}<q$, for which an analogous statement to Proposition 2.5 is simply false without any additional assumptions (counter examples are easy to construct, as in the Introduction). Our first ingredient in the proof of Theorem 2.4 states that our convexity assumptions already suffice to extend Proposition 2.5 to the case $q^{\prime}<q$, $p^{\prime}<p$:

Let $0<p\leq\infty$, $1\leq q\leq\infty$, and set $r=1+\frac{1}{p}-\frac{1}{q}$. Assume that $\frac{1}{2}\leq r\leq 2$. Then under our smooth convexity assumptions, the following statements are equivalent:

where the best constants $D_{p,q}$ and $D^{\prime}_{r}$ above satisfy:

for some universal constants $c_{1},c_{2}>0$. In fact, the direction $(2)\Rightarrow(1)$ holds for $p\geq q$ without any convexity assumptions.

Note that when $p=q=2$, the direction $(2)\Rightarrow(1)$ reduces (up to constants) to Theorem 1.1 (Maz’ya–Cheeger inequality), and the direction $(1)\Rightarrow(2)$ to the Buser–Ledoux Theorems 1.3,1.4. A generalization of Theorem 2.9 involving general Orlicz norms will be derived in .

Let $0<r\leq 1$. Without any convexity assumptions, the $(1/r,1)$ Poincaré inequality:

is equivalent to the following isoperimetric inequality:

The proof of $(2)\Rightarrow(1)$ is thus complete.

Before proceeding to the proof of the direction $(1)\Rightarrow(2)$ (this will be the focus of the next section), let us recall how Theorem 2.9 coupled with Theorem 1.8 conclude the proof of Theorem 2.4 and hence of our Main Theorem 1.5:

By an approximation argument we develop in Section 6, it is enough to prove the theorem under our smooth convexity assumptions.

By Jensen’s inequality, $D_{1,\infty}\geq D_{p,q}$ in the range $1\leq p\leq q\leq\infty$. Employing our (smooth) convexity assumptions, the direction $(1)\Rightarrow(2)$ of Theorem 2.9 implies:

Using our (smooth) convexity assumptions for the second time, Theorem 1.8 asserts that $I$ is concave on $(0,1)$. Since $I$ is also symmetric about $1/2$ (see Corollary 6.5), we immediately deduce that:

which is exactly Cheeger’s isoperimetric inequality, and is identical to stating $D_{1,1}\geq\frac{c}{2}D_{1,\infty}$. Using Proposition 2.5 and Jensen’s inequality if necessary, we can pass from this to an arbitrary $(p^{\prime},q^{\prime})$ inequality in the range $1\leq p^{\prime}\leq q^{\prime}\leq\infty$. ∎

Note that when $\mu$ is the uniform measure on $\Omega$, Theorem 1.8 in fact ensures that $I^{\frac{n}{n-1}}$ is concave, so we may deduce from (2.5) that in fact:

Proposition 2.10 implies that the latter isoperimetric inequality is equivalent to a $(\frac{n}{n-1},1)$ Poincaré inequality. Hence, it is clear that in this case, both our Main Theorem 1.5 and Theorem 2.4 can be strengthened.

The Semi-Group Argument

In this section, we prove the direction $(1)\Rightarrow(2)$ of Theorem 2.9. Our proof closely follows Ledoux’s proof of Theorem 1.4.

Given a smooth complete oriented connected Riemannian manifold $\Omega=(M,g)$ equipped with a probability measure $\mu$ with density $d\mu=\exp(-\psi)dvol_{M}$, $\psi\in C^{2}(M)$, we define the associated Laplacian $\Delta_{(\Omega,\mu)}$ by:

where $\Delta_{\Omega}$ is the usual Laplace-Beltrami operator on $\Omega$. $\Delta_{(\Omega,\mu)}$ acts on $\mathcal{B}(\Omega)$, the space of bounded smooth real-valued functions on $\Omega$. Let $(P_{t})_{t\geq 0}$ denote the semi-group associated to the diffusion process with infinitesimal generator $\Delta_{(\Omega,\mu)}$ (cf. ), characterized by the following system of second order differential equations:

For each $t\geq 0$, $P_{t}:\mathcal{B}(\Omega)\rightarrow\mathcal{B}(\Omega)$ is a bounded linear operator and its action naturally extends to the entire $L_{p}(\mu)$ spaces ($p\geq 1$). We collect several elementary properties of these operators:

$\left|P_{t}(f)\right|^{p}\leq P_{t}(\left|f\right|^{p})$ for all $p\geq 1$.

The following crucial dimension-free reverse Poincaré inequality was shown by Bakry and Ledoux in [5, Lemma 4.2], extending Ledoux’s approach for proving Buser’s Theorem (see also [5, Lemma 2.4], [56, Lemma 5.1]). It may also be interpreted as a weak, dimension-free, form of the Li–Yau parabolic gradient inequality .

Assume that the following Bakry-Émery Curvature-Dimension condition holds on $\Omega$:

Then for any $t\geq 0$ and $f\in\mathcal{B}(\Omega)$, we have:

In fact, the proof of this lemma is very general and extends to the abstract framework of diffusion generators, as developed by Bakry and Émery . We comment that in the Riemannian setting, it is known (see also ) that the gradient estimate of Lemma 3.1 is preserved when restricting to a locally convex domain (as defined in the Appendix) with smooth boundary; we refer to Sturm [83, Proposition 4.15] for a general statement about closedness of the Bakry-Émery Curvature-Dimension condition in an arbitrary metric probability space. The above lemma therefore holds under more general conditions, namely when $\mu$ is supported on a locally convex domain $\Omega\subset(M,g)$ with $C^{2}$ boundary, and $d\mu|_{\Omega}=\exp(-\psi)dvol_{M}|_{\Omega}$, $\psi\in C^{2}(\overline{\Omega})$. In this case, $\Delta_{\Omega}$ in (3.1) denotes the Neumann Laplacian on $\Omega$, $\mathcal{B}(\Omega)$ denotes the space of bounded smooth real-valued functions on $\Omega$ satisfying Neumann’s boundary condition on $\partial\Omega$, and Lemma 3.1 remains valid.

Our convexity assumptions are that $K=0$ in Lemma 3.1, and this is what we will henceforth assume. It is clear that our results in this section may be extended to the case of $K>0$, but we do not pursue this direction in this work.

From Lemma 3.1, it is immediate that for any $2\leq q\leq\infty$:

and using $q=\infty$, Ledoux easily deduces the following dual statement [56, (5.5)]:

First, our assumption on the range of $r$ implies that by applying Proposition 2.5 if necessary, we may assume that $p\geq 1,q\geq 2$ at the expense of an additional universal constant appearing in (2.4). An additional universal constant will appear on account of Lemma 2.1, with which we pass to $E_{\mu}$ instead of $M_{\mu}$ in (1), so our assumption now reads:

Let $A$ denote an arbitrary Borel set in $\Omega$, and let $\chi_{A,\varepsilon}(x):=(1-\frac{1}{\varepsilon}d_{g}(x,A))\vee 0$ denote a continuous approximation in $\Omega$ to the characteristic function $\chi_{A}$ of $A$. Clearly:

Applying Corollary 3.2 to functions in $\mathcal{B}(\Omega)$ which approximate $\chi_{A,\varepsilon}$ (in say $W^{1,1}(\Omega,\mu)$) and passing to the limit inferior as $\varepsilon\rightarrow 0$, it follows that:

We start by rewriting the right hand side above as:

Note that by Hölder’s inequality (recall that $p\geq 1$) and our assumption (3.5):

Using (3.3) (recall that $q\geq 2$) to estimate $\left\|\left|\nabla P_{t}(\chi_{A})\right|\right\|_{L_{q}(\mu)}$, we conclude that:

We may now optimize on $t$. Using the rough estimate:

for $s\geq 1$, we evaluate (3.6) at time:

where $r=2-1/q-1/p^{*}=1+1/p-1/q$. Since $r\leq 2$, this concludes the proof. ∎

As evident from the proof, for deducing the direction $(1)\Rightarrow(2)$ of Theorem 2.9, the definition of smooth convexity assumptions given in the Introduction may be extended to encompass the more general case treated in this section. Moreover, it is possible to provide an approximation argument for deducing this direction without any smoothness assumptions. We provide the argument in and omit it here, since it is not required for the results of this work.

Interpretations and Extensions

In this section, we provide some further interpretations and extensions of our Main Theorem, which will also be needed for the applications of the next section. We assume throughout this section that our convexity assumptions on $(\Omega,d,\mu)$ are satisfied.

Lemma 2.2 demonstrates that if $A$ is a set with $\mu(A)\leq 1/2$ on which the minimal ratio $D_{Che}=\mu^{+}(A)/\mu(A)$ in Cheeger’s isoperimetric inequality is attained (or nearly attained), then the function $f=\chi_{A}$ (or the sequence of Lipschitz functions which approximate it) attains the same (nearly) minimal ratio

among all functions $f\in\mathcal{F}$ with $M_{\mu}f=0$. Clearly $\chi_{A}$ (or its approximating sequence) is far from being $1$-Lipschitz. If on the other hand we define:

which is a $1$-Lipschitz function, it is not clear that it will have a small ratio in (4.1). Our Main Theorem 1.5 (together with Lemma 2.1) states that under our convexity assumptions, any $1$-Lipschitz function $f_{0}$ on $(\Omega,d)$ with $M_{\mu}f_{0}=0$ which is (essentially) optimal in the First-Moment inequality (say $\int|f_{0}|d\mu\geq 1/(3D^{M}_{FM})$), also essentially minimizes the ratio in (4.1). Moreover, using the co-area formula as in Lemma 2.2 and applying our Main Theorem, we have:

from which we also see that the ratio $\mu^{+}(A_{t})/\min(\mu(A_{t}),1-\mu(A_{t}))$ for the “average” level set $A_{t}$ of $f_{0}$ is essentially $D_{Che}$, the smallest possible.

Theorem 1.6 from the Introduction states that $f_{0}$ as above may in fact be chosen to be of the form (4.2).

Given a Borel set $A\subset\Omega$ with $\mu(A)\geq 1/2$, we denote $g_{A}(x)=d(x,A)$. Clearly $g_{A}$ is $1$-Lipschitz and $M_{\mu}g_{A}=0$, so one direction follows immediately by Lemma 2.2:

For the other direction, we employ our Main Theorem (and Lemma 2.1):

where the infimum is over all $1$-Lipschitz functions $f$ on $(\Omega,d)$ with $M_{\mu}f=0$. Denoting $A_{1}=\left\{f\leq 0\right\},A_{2}=\left\{f\geq 0\right\}$, we have $\mu(A_{i})\geq 1/2$, $i=1,2$. By continuity of $f$, $f|_{\partial A_{1}}\equiv 0$, $f|_{\partial A_{2}}\equiv 0$ (even though it is possible that $\partial A_{1}\neq\partial A_{2}$), and since it is $1$-Lipschitz:

The next proposition will prove to be very useful for the applications of the next section. We start with some notations. Given a Borel function $f$ on a Borel probability space $(\Omega,\mu)$ and $\delta\in$, let us denote by $Q_{\delta}(f)=Q_{\mu,\delta}(f)$ the $\delta$-quantile of $f$:

Let us also recall an inequality due to Paley and Zygmund (see also [46, Chapter 2]), which in its simplest form reads as follows:

Let $f$ denote a Borel function on $\Omega$, and assume that:

Then for any $\theta\in(0,1)$, denoting $\varepsilon(\theta)=(1-\theta)^{2}/D^{2}$, one has $Q_{1-\varepsilon(\theta)}(|f|)\geq\theta\left\|f\right\|_{L_{1}(\mu)}$.

Let $f_{0}$ denote a $1$-Lipschitz function with either $M_{\mu}f_{0}=0$ and $\left\|f_{0}\right\|_{L_{1}(\mu)}\geq 1/(2D^{M}_{FM})$ or $E_{\mu}f_{0}=0$ and $\left\|f_{0}\right\|_{L_{1}(\mu)}\geq 1/(2D_{FM})$. Then:

for some universal constants $C_{0}>0$ and $0<\varepsilon_{0}<1$.

Proceeding as in Corollary 2.7, and using Lemma 2.1 and the Main Theorem:

for some universal constant $D_{0}>0$, and (4.4) follows by Lemma 4.1 (with $\theta=1/2$). Note that our convexity assumptions necessarily imply that $\left\|f_{0}\right\|_{L_{1}(\mu)}<\infty$ (see Lemma 6.13), so the appeal to Lemma 4.1 is indeed legitimate. ∎

An arbitrarily slow uniform tail decay condition (1.3) implies any of the statements of the Main Theorem 1.5, with $D_{Che},D_{Poin},D_{Exp},D_{FM}$ depending solely on $\alpha$. Moreover, $E_{\mu}f$ in (1.3) may be replaced by $M_{\mu}f$.

Given a $1$-Lipschitz function $f_{0}$ satisfying either of the assumptions of Proposition 4.2, these and (4.4) imply that:

Consequently, the tail decay condition (1.3) (whether stated with $E_{\mu}f$ or $M_{\mu}f$) ensures that $\max(D_{FM},D^{M}_{FM})\geq 1/(4\alpha^{-1}(\varepsilon_{0}))>0$, so by Lemma 2.1 the First-Moment concentration inequality is satisfied, from which the other statements of the Main Theorem follow. ∎

Applications to Spectral Gap of Convex Domains

First, we would like to obtain a stability result for $D_{Che}(\Omega)$ (or equivalently $D_{Poin}(\Omega)$) for perturbations of $\Omega$. Clearly, without any further assumptions, there can be no such result (as seen by adding arbitrarily small “necks” to $\Omega$ as in the Introduction), so we restrict our attention to convex domains. In this case, our Main Theorem 1.5 asserts that this is equivalent to obtaining a stability result for $D_{FM}(\Omega)$, which is much easier. To obtain the best quantitative bounds, we will also employ $D_{Exp}(\Omega)$.

Let $f_{0}$ denote a $1$-Lipschitz function on $L$ with $M_{\lambda_{L}}f_{0}=0$ so that $\int|f_{0}|d\lambda_{L}\geq 1/(2D^{M}_{FM}(L))$. Since $L$ is convex, we may clearly extend $f_{0}$ to a $1$-Lipschitz function on $K$, say by defining $f_{1}=f_{0}(\text{Proj}_{L}x)$. Here $\text{Proj}_{L}x$ denotes the unique (by convexity) $y$ in $L$ so that $d(x,L)=d(x,y)$. We may assume that $E_{\lambda_{K}}f_{1}\geq 0$ (otherwise exchange $f_{0}$ with $-f_{0}$). Note that we can estimate $E_{\lambda_{K}}f_{1}$ as follows:

By Proposition 4.2, there exists some universal $\varepsilon_{0}>0$ so that $\left\|f_{0}\right\|_{L_{1}(\lambda_{L})}\leq Q_{\lambda_{L},1-\varepsilon_{0}}(|f_{0}|)$. Using this, the ratio between the volumes of $L$ and $K$, the triangle inequality, the Markov-Chebyshev inequality and the estimate on $E_{\lambda_{K}}f_{1}$ in (5.1), we evaluate:

where $C_{0}>0$ is some universal constant. Using Lemma 2.1 and (2.3), the assertion follows. ∎

Note that for any $1/2<p\leq 1$ and in fact even without assuming that $L$ is convex:

Indeed, since $K$ is convex, by Theorem 1.8 (more precisely, its extension to non-smooth domains or densities given by Theorem 6.10 and Corollaries 6.11,6.12) we know that $D_{Che}(K)=2I_{(K,\left|\cdot\right|,\lambda_{K})}(1/2)$. Given a Borel set $A$ with $\lambda_{K}(A)=1/2$, we have:

By the assumption in (5.2), $1-\frac{1}{2p}\leq\lambda_{L}(A)\leq\frac{1}{2p}$, and from this we easily deduce the conclusion in (5.2). Iterating this using a sequence of intermediate convex bodies (here we already need to use that $L$ is convex) $L=L_{0}\subset L_{1}\subset\ldots\subset L_{m}=K$ so that $\textnormal{Vol}\left(L_{i}\right)/\textnormal{Vol}\left(L_{i+1}\right)\geq v^{1/m}>1/2$ (for example, assuming $0\in L$, choose $L_{i}=(1+r_{i})L\cap K$ for appropriate $r_{i}\geq 0$), we obtain that:

Taking the limit as $m\rightarrow\infty$ yields the claimed power of $2$ (even without any additional numerical constant!). ∎

then applying Lemma 5.2, the Main Theorem 1.5 and Lemma 5.1, we obtain:

for some universal constants $c_{i}>0$, concluding the proof of Theorem 1.7. Of course a similar upper bound on $D_{Che}(K)$ is obtained by interchanging the roles of $K,L$.

In Convexity Theory, many interesting ways are known to cut a convex body $K$ so that its volume is preserved up to a constant (e.g. by slabs, parallelepipeds, balls etc…). We see that all of these preserve (up to a constant) $D_{Che}(K)$ (equivalently, the spectral gap $D_{Poin}^{2}(K)$). A useful way to measure the distance between two convex bodies is given by the following variant on the usual geometric distance:

so by passing from the outer to the inner body (in which case our estimates are logarithmic), we deduce:

for some $1\leq s\leq C_{1}n$, where $C_{1}>0$ is some universal constant, then:

where $C_{2}>0$ is another universal constant.

Denoting $a,b$ the best constants in (5.4) and applying Lemma 5.1:

and since $b\leq d_{G}(K,L)\leq C_{1}+1$, the assertion follows. ∎

appearing in the assumptions of both lemmas has a clear and intuitive geometric meaning.

Lemmas 5.1 and 5.2 remain valid for absolutely continuous log-concave probability measures $\mu_{K},\mu_{L}$ (replacing respectively $K,L$), if the condition (5.5) in the assumption is replaced by the condition:

and $D_{Che}(\Omega),D_{FM}(\Omega),D_{Exp}(\Omega)$ are replaced by $D_{Che}(\mu_{\Omega}),D_{FM}(\mu_{\Omega}),D_{Exp}(\mu_{\Omega})$ ($\Omega=K,L$) in the corresponding conclusion.

Identical to the proof of the original lemmas; the only minor point is the construction of intermediate measures $\mu_{L_{i}}$ in the proof of Lemma 5.2, which may be defined e.g. by $\mu_{L_{i}}=\frac{\eta_{L_{i}}}{|\eta_{L_{i}}|}$, $\frac{d\eta_{L_{i}}}{dx}(x)=\min((1+r_{i})\frac{d\mu_{L}}{dx}(\frac{x}{1+r_{i}}),\frac{d\mu_{K}}{dx}(x))$, for appropriate $r_{i}>0$ (assuming the origin is in the interior of the support of $\mu_{L}$). ∎

The analogue of Theorem 1.7 may then be conveniently formulated using the total-variation metric:

with $c(\varepsilon)=c^{\prime}\varepsilon^{2}/\log(1+1/\varepsilon)$ and $c^{\prime}>0$ a universal constant.

Let $\mu_{0}$ denote the measure whose density is $\min(\frac{d\mu_{1}}{dx},\frac{d\mu_{2}}{dx})$, and note that $d_{TV}(\mu_{1},\mu_{2})=1-|\mu_{0}|$. Denoting by $\mu_{3}$ the (log-concave) probability measure $\frac{\mu_{0}}{|\mu_{0}|}$, since $\frac{d\mu_{i}}{dx}\geq|\mu_{0}|\frac{d\mu_{3}}{dx}$, $i=1,2$, we may apply Lemma 5.4 and the Main Theorem to pass from $\mu_{1}$ to $\mu_{3}$ to $\mu_{2}$ as in (5.3), concluding the proof. ∎

2 Optimality of Stability

To the best of our knowledge, no quantitative results on the stability of $D_{Che}$ or $D_{Poin}$ for convex domains with respect to volume preserving perturbations or geometric distance were previously known. Moreover, we claim that the bounds obtained in Theorem 1.7 (or (5.3)) are optimal (up to numeric constants) with respect to $v_{L}$ and close to optimal with respect to $v_{K}$ (note that the dependence is logarithmic in the former yet quadratic in the latter; in other words, the deterioration in the Cheeger constant when passing from an outer convex body to an inner one is genuinely different than when passing from the inner one outward). This is witnessed by the following:

It is known (see Subsection 5.5) that $D_{Che}(Q^{m})\simeq D_{Che}(B_{1}^{m})\simeq 1$, so by the $-1$-homogeneity of $D_{Che}$, it follows that $D_{Che}(K_{k})\simeq 1$ and $D_{Che}(L_{k})\simeq\frac{1}{k}$. Denoting $v_{k}=\frac{\textnormal{Vol}\left(L_{k}\right)}{\textnormal{Vol}\left(K_{k}\right)}$, since $\log 1/v_{k}\simeq k$, we conclude that:

uniformly for all $k=2,\ldots,n-1$. So one cannot expect better than logarithmic dependence on $1/v$ (at least when $v\geq\exp(-n)$), which coincides with the estimate given by Lemma 5.1.

On the other hand (as is well-known), if we set $L=Q^{n}$ and $K=Q^{n-1}\times tQ^{1}$ a circumscribing box with $t>1$, since $D_{Che}(K)\simeq 1/t$ in that range, it is clear that the quadratic dependence on $v$ in Lemma 5.2 cannot be improved beyond linear. Although we do not know whether the optimal bound is, up to a constant, closer to the linear or quadratic asymptotic, we comment that for very small perturbations (i.e. $v$ very close to 1), it is possible to show that the exact quadratic bound in Lemma 5.2 is optimal (in this range of $v$, we of course do not allow any additional numerical constants).

The next example (which is similar yet different from the previous one) shows that the bounds in Corollary 5.3 are optimal too (up to numeric constants), as a function of $s$ in the stated range.

Since clearly $L_{s}\subset K_{s}$, it remains to note that $(1-\frac{s}{r_{n}})K_{s}\subset L_{s}$, so $d_{G}(K_{s},L_{s})-1\simeq\frac{s}{n}$. By interchanging the roles of $K_{s},L_{s}$ appropriately, we observe that the estimates on $D_{Che}(K)/D_{Che}(L)$ in Corollary 5.3 are sharp both from above and from below.

It is easy to adapt the proofs of Lemma 5.1 and consequently Corollary 5.3 to obtain even sharper quantitative bounds (up to universal constants) on the stability of $D_{Che}$ for specific convex bodies, such as the Euclidean ball $B_{2}^{n}$. For instance, in the latter case, one obtains that if $d_{G}(K,B_{2}^{n})\leq 1+\frac{s}{n}$ for $1\leq s\leq C_{1}n$, then:

This is an improvement over Corollary 5.3 and known to be sharp for $s=n$ (folklore).

It is well known and immediate to see that isoperimetric inequalities are preserved under $1$-Lipschitz mappings. Given two metric probability spaces $(X,d_{X},\mu)$ and $(Y,d_{Y},\nu)$, recall that a Borel map $T:(X,d_{X})\rightarrow(Y,d_{Y})$ is said to push forward $\mu$ onto $\nu$, if $\nu(A)=\mu(T^{-1}(A))$ for every Borel set $A\subset Y$. This is equivalent to requiring that for any Borel function $g$ on $(Y,d_{Y})$:

This will be denoted by $T_{*}(\mu)=\nu$. The following is then immediate from the definitions:

The following result states that when our convexity assumptions hold for the target space, as far as Cheeger’s isoperimetric inequality is concerned, one need not require that $T$ be Lipschitz on the entire space, but rather just on average. We would like to thank Bo’az Klartag for a fruitful discussion regarding this point.

Assume that $(Y,d_{Y},\nu)$ verifies our convexity assumptions and that $T_{*}(\mu)=\nu$ for some Lipschitz-on-balls map $T$. Then:

Here $\left\|DT\right\|_{op}(x)$ denotes the local Lipschitz constant of $T$ at $x$:

When $T$ is smooth and $X,Y$ are linear spaces, this coincides with the operator norm of the usual derivative matrix $DT$ at $x$.

First, rewrite Cheeger’s isoperimetric inequality on $(X,d_{X},\mu)$ in functional form (Lemma 2.2):

Using this, we estimate the First-Moment constant on $(Y,d_{Y},\nu)$. Given a $1$-Lipschitz function $g$ on $(Y,d_{Y})$, clearly $g\circ T$ is Lipschitz-on-balls on $(X,d_{X})$, hence in $\mathcal{F}(X,d_{X})$. We then have by the definition of push-forward and our assumption (5.6):

We conclude by our Main Theorem (and Lemma 2.1), which imply that $D_{Che}(Y,d_{Y},\nu)\geq cD^{M}_{FM}(Y,d_{Y},\nu)$ under our convexity assumptions on $(Y,d_{Y},\nu)$. ∎

In this subsection, we easily recover some previously known estimates on the Cheeger constant of convex domains in a single framework and extend some results to the Riemannian setting. We begin with the following stimulating conjecture from :

Here $\sigma_{1}(\mu)^{2}$ denotes the largest eigenvalue of the symmetric covariance matrix $\Sigma(\mu)$ of $\mu$:

We will write $\sigma_{1}(K)$ for $\sigma_{1}(\lambda_{K})$.

Although the KLS conjecture is far from being resolved, some general lower bounds on $D_{Che}$ are known, but these produce dimension-dependent results. We will see that our Main Theorem easily reproduces these bounds.

The following result in the Euclidean setting is due to Payne and Weinberger . This was generalized to the Riemannian setting by Li and Yau . We refer to the Appendix for missing definitions.

If $K\subset(M,g)$ is a locally convex bounded domain with smooth boundary and $Ric_{g}\geq 0$, then:

where diam denotes the diameter and $d_{g}$ the induced geodesic distance. In fact, when $(M,g)$ is Euclidean space the constant 2 above may be omitted.

Ledoux’s Theorem 1.4 implies that the same lower bound (up to an additional constant) holds for $D_{Che}(K,d_{g},\lambda_{K})$. In the Euclidean case, this was strengthened in :

To obtain this result, Kannan, Lovász and Simonovits developed a geometric localization technique (which in fact can be traced back to the work of M. Gromov and V. Milman ). As pointed out to us by Sasha Sodin, it is interesting to note that this technique uses some geometric properties of Euclidean space and does not generalize to other Riemannian manifolds (except in special cases, like that of the Euclidean Sphere, as in the work of Gromov–Milman). Our method, on the other hand, does allow us to state the following generalization of Theorem 5.11 to the Riemannian setting, which also improves over Theorem 5.10:

Assume that $(\Omega,d,\mu)$ satisfies our convexity assumptions. Then:

As usual, we just need to bound $D_{FM}(\Omega,d,\mu)$. Let $f$ denote a $1$-Lipschitz function on $(\Omega,d)$. Then for any $x_{0}\in\Omega$, applying the triangle inequality twice:

and the claim follows by our Main Theorem. ∎

An alternative approach to localization for proving isoperimetric inequalities was developed by Bobkov in the Euclidean setting. Bobkov’s approach was extended by Barthe and subsequently by Barthe and Kolesnikov . This approach is based on the Prékopa–Leindler inequality (e.g. ), or equivalently, on optimal transportation, which have both been recently generalized to the Riemannian-with-density-setting by Cordero-Erausquin, McCann and Schmuckenschläger . Using these tools we expect that it should be possible to provide an alternative proof of Theorem 5.12 following Bobkov’s approach, but as pointed out to us by one of the referees, this has yet to be accomplished. We would like to thank the referee for his comments regarding our original simpleminded remark in this direction.

We would like to mention another bound on $D_{Che}$ obtained in using the localization method.

where $\theta_{B}(x)$ denotes the longest symmetric interval contained in $B$ and centered at $x$, and $c>0$ is a universal constant.

We have recently managed to derive this result using our Main Theorem, but this will be described elsewhere. Instead, we would like to show how this bound may be used to recover a result of Bobkov ; in fact, the bound we deduce is formally stronger than Bobkov’s. Bobkov employs the localization method as well, but then relies on some nice trick involving moment inequalities for polynomials in the log-concave setting. Our argument, on the other hand, is more geometric. Independently of our proof, we heard about a similar idea for bounding the boundary measure of large sets from Santosh Vempala (using localization as well).

where $Var_{\mu}$ denotes the variance with respect to $\mu$.

Without loss of generality, we may assume that $x_{0}=0$; for general $x_{0}$ the claimed bound follows by translating $\mu$. Let $E:=E_{\mu}|x|$, $S:=(Var_{\mu}|x|)^{1/2}$, and denote:

By Chebyshev’s inequality, $\mu(B)\geq 3/4$, so if we define $\mu_{0}:=\mu|_{B}/\mu(B)$, it follows that $d_{TV}(\mu,\mu_{0})\leq 1/4$. Hence $D_{Che}(\mu)\simeq D_{Che}(\mu_{0})$ by Theorem 5.5. Assume that $E\geq 2S$, otherwise the support of $\mu_{0}$ has diameter bounded by $8S$, and one can conclude as in Theorem 5.12. We now employ Theorem 5.14 to bound $D_{Che}(\mu_{0})$:

The crucial geometric observation is that for the Euclidean ball $B$:

It remains to plug this into (5.8) and evaluate the resulting expression using integration by parts and Chebyshev’s inequality. We leave it as an exercise to conclude that:

for some universal constant $c^{\prime}>0$. This bound is in fact formally better than Bobkov’s bound (by several applications of Hölder’s inequality), but using some standard results in Convexity Theory, it is in fact equivalent in the interesting situations. ∎

This is immediate from the results of Schechtman and Zinn , who showed that $D_{Exp}$ of these bodies is bounded from below by a universal constant. The result then follows from our Main Theorem (in fact, we only need a bound on $D_{FM}$). ∎

Another family of convex bodies for which the KLS conjecture is almost confirmed, is that of unconditional convex bodies $K$, i.e. convex bodies for which $(x_{1},\ldots,x_{n})\in K$ iff $(\pm x_{1},\ldots,\pm x_{n})\in K$. It was recently shown by Bo’az Klartag that if $K$ is an unconditional body with $\sigma_{1}(K)=1$ then $D_{Che}(K)\geq c/\log n$, for some universal constant $c>0$. To obtain this result, Klartag employed Theorem 1.7 to pass to an unconditional body contained inside the cube $(C\log n)^{n}$, and then used some symmetry properties of the Laplacian’s eigenfunctions to conclude his result. In fact, one can just use Theorem 1.8 on the concavity of the isoperimetric profile (in the form of Lemma 5.2) for this application.

We conclude this section by stating the known dimension dependent bounds on $D_{Che}(K)$ for non-degenerate convex bodies $K$ (in the sense that $\sigma_{1}(K)=1$).

It is known in this case that $\textnormal{diam}(K)\leq cn$ (by a simple volume estimate). Theorem 5.10 (together with Theorem 1.4) then gives $D_{Che}(K)\geq c/n$. The first KLS bound (Theorem 5.11) improves this to $D_{Che}(K)\geq c/\sqrt{n}$, since:

Bobkov’s bound (Theorem 5.15) is always at least as good as the first KLS bound (up to a constant), since (using the bound derived in the proof together with a standard application of Borell’s lemma ):

for some universal constant $C>0$. We see that whenever some non-trivial information on $Var_{\mu}(\left|x-x_{0}\right|)$ is known, Bobkov’s bound is strictly better. Such a remarkable result was proved by Bo’az Klartag , allowing him to deduce a Central-Limit type result for the class of convex bodies (and more generally, log-concave measures). Klartag’s improved estimate in reads:

Approximation Argument

In this section, we develop an approximation argument for extending the following theorems to non-necessarily smooth densities (or boundaries) in our convexity assumptions:

Theorem 1.8 on the concavity of the isoperimetric profile.

We will develop different procedures for extending each of these theorems.

We begin by extending our definition of smooth convexity assumptions (we refer to the Appendix for the definition of locally convex).

We will say that our generalized smooth convexity assumptions are fulfilled if:

$\Omega\subset M$ is a locally convex domain with $C^{2}$ boundary.

$d$ denotes the induced geodesic distance on $(M,g)$.

$d\mu=\exp(-\psi)dvol_{M}|_{\Omega}$, $\psi\in C^{2}(\overline{\Omega})$, and as tensor fields on $\Omega$:

This definition was already used in the statement of Theorem 1.8 on the concavity of the isoperimetric profile. The smoothness assumptions in the above definition are used in an essential way in the proof of this theorem to deduce the existence and regularity of the isoperimetric minimizers, which are otherwise false. This permits the use of variational methods from Riemannian Geometry, consequently obtaining a second-order differential inequality which the isoperimetric profile must satisfy (see the Appendix for more details). Nevertheless, the restriction to smooth densities and domains still seems like a technical artifact of the proofs. Some authors have suggested various methods to remove these smoothness assumptions (see e.g. Morgan and Bayle [12, Chapter 4]), but unfortunately these are not well suited for our purposes. We therefore attempt to use a different approximation argument for extending Theorem 1.8 to a more general setting.

At first glance, it is tempting to believe that the isoperimetric profile of $(\Omega,d,\mu)$ should be stable under approximating the measure $\mu$ by measures $\mu_{m}$ in, say, total-variation distance. However, the profile is in fact not even pointwise continuous under arbitrary approximation in total-variation. To see this, consider the measures $\mu_{m}$ which are uniform on the set $\setminus[1/2-1/m,1/2+1/m]$, and converge to $\mu$, the uniform measure on $$. Clearly$I_{\mu_{m}}(1/2)=0$for every$m\geq 3$, even though$I_{\mu}(1/2)=1$. So one must take care when specifying the approximation.