Remarks on the KLS conjecture and Hardy-type inequalities

Introduction

The classical Faber–Krahn inequality (e.g. ) states that under the same conditions:

where $P^{D}_{\Omega^{*}}$ is the best constant in the above inequality under the same conditions with $\Omega=\Omega^{*}$, the Euclidean Ball having the same volume as $\Omega$. $P^{D}$ is called the Poincaré constant with zero Dirichlet boundary conditions; it is elementary to verify that $P^{D}_{\Omega^{*}}\simeq\frac{1}{n}\left|\Omega^{*}\right|^{2/n}$ (see Remark 3 for more precise information).

In this note we explore what may be said when $f$ does not necessarily vanish on the boundary, and develop applications for estimating the Poincaré constant with Neumann boundary conditions. Here and elsewhere, we use $\left|M\right|$ to denote the $k$-dimensional Hausdorff measure $\mathcal{H}^{k}$ of the $k$-dimensional manifold $M$, and $A\simeq B$ to denote that $c\leq A/B\leq C$, for some universal numeric constants $c,C>0$. All constants $c,c^{\prime},C,C^{\prime},C_{1},C_{2}$, etc. appearing in this work are positive and universal, i.e. do not depend on $\Omega$, $n$ or any other parameter, and their value may change from one occurrence to the next.

Let $\lambda_{\Omega}$ denote the uniform (Lebesgue) probability measure on $\Omega$, and let $P^{N}_{\Omega}$ denote the Poincaré constant of $\Omega$, i.e. the best constant satisfying:

It is easy to reduce the KLS conjecture to the case that $\Omega$ is isotropic, meaning that its barycenter is at the origin and the variance of all unit linear functionals is $1$, i.e.:

Set $P^{\infty}_{\Omega}:=P^{\infty}_{\lambda_{\Omega}}$; clearly $P^{\infty}_{\Omega}\leq P^{N}_{\Omega}$. In , the second-named author showed that when $\Omega$ is convex, the latter inequality may be reversed:

In this work, we obtain several additional reductions of the KLS conjecture. First, we obtain a sufficient condition by reducing to the study of $P^{\infty}_{\sigma_{\partial\Omega}}$ and $P^{\infty}_{\lambda_{\partial\Omega}}$, the cone and Lebesgue measures on $\partial\Omega$, respectively. In particular, it suffices to bound the variance of homogeneous functions which are $1$-Lipschitz on the boundary. This is achieved by obtaining Neumann versions of the Hardy and Faber-Krahn inequalities (1) and (2) for general functions (not necessarily vanishing on the boundary). The parameters $P^{\infty}_{\sigma_{\partial\Omega}}$ or $P^{\infty}_{\lambda_{\partial\Omega}}$ may then be bounded using a result from our previous work , by averaging certain curvatures on $\partial\Omega$ (see Theorem 11).

Second, we reduce the KLS conjecture to the class of harmonic functions. Thirdly, we consider the Poincaré constant of an unconditional convex body bounded by the principle hyperplanes, when a certain mixed Dirichlet–Neumann boundary condition is imposed. It is interesting to check which of the boundary conditions will dominate this Poincaré constant, and we determine that it is the Dirichlet ones, resulting in a Faber–Krahn / Hardy-type upper bound.

Our proofs follow classical arguments for establishing the Hardy inequality, which can be viewed as a Lyapunov function or vector-field method, in which one is searching for a vector-field whose magnitude is bounded from above on one hand, and whose divergence is bounded from below on the other. For more applications of Lyapunov functions to the study of Sobolev-type inequalities, see .

Acknowledgements. We would like to thank Bo’az Klartag for his interest and fruitfull discussions. The first-named author was supported by RFBR project 12-01-33009 and the DFG project CRC 701. This study (research grant No 14-01-0056) was supported by The National Research University -Higher School of Economics’ Academic Fund Program in 2014/2015. The second-named author was supported by ISF (grant no. 900/10), BSF (grant no. 2010288), Marie-Curie Actions (grant no. PCIG10-GA-2011-304066) and the E. and J. Bishop Research Fund.

Hardy-type inequalities

Applying this to $g=f^{2}$ and using the Cauchy-Schwartz inequality (in additive form), we obtain for any positive function $\lambda$ on $\Omega$:

Let us apply this to several different vector fields $\xi$.

In this subsection, assume in addition that $\Omega$ is star-shaped, meaning that $\Omega=\left\{x\;;\;\left\|x\right\|\leq 1\right\}$, where $\left\|x\right\|:=\inf\left\{\lambda>0;x\in\lambda\Omega\right\}$ denotes its associated gauge function. We denote by $\sigma_{\partial\Omega}$ the induced cone probability measure on $\partial\Omega$, i.e. the push-forward of $\lambda_{\Omega}$ via the map $x\mapsto\frac{x}{\left\|x\right\|}$. It is well-known and immediate to check that:

Let $f$ denote a smooth function on $\Omega$. Then:

Apply (7) with $\xi(x)=x$, so that $div(\xi)=n$, and $\lambda\equiv n/2$. We obtain:

In particular, we see that (8) immediately follows when $f$ vanishes on $\partial\Omega$. For general functions, we divide (9) by $Vol(\Omega)$ and apply the resulting inequality to $f-a$ with $a:=\int_{\partial\Omega}fd\sigma_{\partial\Omega}$:

2 Optimal Transport to Euclidean Ball

Let $f$ denote a smooth function on $\Omega$. Then:

Applying (7) with $\lambda\equiv\alpha/2$, and using that $\xi=\nabla\varphi_{0}\in B_{2}^{n}$, we obtain:

In particular, when $f$ vanishes on $\partial\Omega$, we deduce (2) with a slightly inferior constant; however, this constant is asymptotically (as $n\rightarrow\infty$) best possible, see Remark 3 below. Dividing by $\left|\Omega\right|$ and applying the resulting inequality to $f-a$ with $a:=\int_{\partial\Omega}fd\lambda_{\partial\Omega}$, the assertion follows. ∎

It is known (e.g. [13, p. 139]) that $1/P^{D}_{B_{2}^{n}}$ is equal to the square of the first positive zero of the Bessel function of order $(n-2)/2$. According to [33, p. 516], the first zero of the Bessel function of order $\beta$ is $\beta+c_{0}\beta^{1/3}+O(1)$, for a constant $c_{0}\simeq 1.855$, and so consequently $P^{D}_{B_{2}^{n}}=\frac{4}{n^{2}}(1+o(1))$. By homogeneity, it follows that $P^{D}_{\Omega^{*}}=4\left|\Omega^{*}\right|^{2/n}/(n^{2}\left|B_{2}^{n}\right|^{2/n})(1+o(1))$, confirming that the constant in Theorem 2 is asymptotically best possible.

Note that if we start from (6) and avoid employing the Cauchy-Schwartz inequality used to derive (7), the above proof (using $\xi=\nabla\varphi_{0}$ and $g\equiv 1$) yields the isoperimetric inequality with sharp constant for smooth bounded domains:

This proof was first noted by McCann , extending an analogous proof by Knothe and subsequently Gromov of the Brunn-Minkowski inequality using the Knothe map . See for rigorous extensions of such an approach to non-smooth domains.

3 Normal vector field

In this subsection, we assume in addition that $\Omega$ is strictly convex. We employ the vector field:

the exterior unit normal-field to the convex set $\Omega_{x}:=\left\|x\right\|\Omega$. Note that this field is not well defined (and in particular not continuous) at the origin, so strictly speaking we cannot appeal to (7). However, this is not an issue, since $div(\xi)$ is homogeneous of degree $-1$, and so the Jacobian term in polar coordinates $r^{n-1}$ will absorb the blow-up of the divergence near the origin (recall $n\geq 2$). To make this rigorous, we simply repeat the derivation of (7) by integrating by parts on $\Omega\setminus\epsilon B_{2}^{n}$, and note that we may take the limit as $\epsilon\rightarrow 0$, since the contribution of the additional boundary $\partial\epsilon B_{2}^{n}$ goes to zero as $\xi$ and $f$ are bounded.

where $H_{S}(y)$ denotes the mean-curvature (trace of the second fundamental form $\text{II}_{S}$) of a smooth oriented hypersurface $S$ at $x$. Indeed, by definition $\nabla\xi|_{\xi^{\perp}}=\text{II}_{\partial\Omega_{x}}$, and $2\nabla_{\xi}\xi=\nabla\left\langle\xi,\xi\right\rangle=0$, and so $div(\xi)=tr(\nabla\xi)=H_{\partial\Omega_{x}}$.

For any strictly convex $\Omega$ and smooth function $f$ defined on it:

Immediate after appealing to (7) with $\lambda(x)=\frac{1}{2}\frac{H_{\partial\Omega}(x/\left\|x\right\|)}{\left\|x\right\|}$. ∎

We note for future reference that by (6) with $g\equiv 1$ we have:

Also, integration in polar coordinates immediately verifies:

4 Unconditional Sets

Finally, we consider one additional vector-field for the Lyapunov method, which is useful when $\Omega$ is the intersection of an unconditional convex set with the first orthant $Q:=[0,\infty)^{n}$ under a certain mixed Dirichlet–Neumann boundary condition. Let $int(Q)$ denote the interior of $Q$.

Let $\Omega\subset Q$ denote a set having smooth boundary, such that every outer normal $\nu$ to $\partial\Omega\cap int(Q)$ has only non-negative coordinates. Let $f$ denote a smooth function vanishing on $\partial Q$. Then:

Since $\left\langle\xi,\nu\right\rangle\leq 0$ in $int(Q)\cap\partial\Omega$ and $f|_{\partial Q\cap\partial\Omega}=0$, we have:

Finally, by the arithmetic-harmonic means inequality, we obtain:

We stress that this result is very similar to the following variant of the Hardy inequality:

which holds for any smooth $f$ vanishing on $\partial\Omega$ (see ).

Reduction of KLS conjecture to subclasses of functions

Let us now see how the Hardy-type inequalities of the previous section may be used to reduce the KLS conjecture to the behaviour of $1$-Lipschitz functions on the boundary $\partial\Omega$. We remark that we use here the term “reduction” in a rather loose sense - we obtain a sufficient condition for the KLS conjecture to hold, but we were unable to show that this is also a necessary one.

Together with (5), Theorem 1 immediately yields:

For any smooth convex domain $\Omega$ with barycenter at the origin:

Apply Theorem 1 to an arbitrary $1$-Lipschitz function $f$, and note that $\int\left|x\right|^{2}d\lambda_{\Omega}=\sum_{i=1}^{n}Var_{\lambda_{\Omega}}(x_{i})\leq nP_{\Omega}^{Lin}$. ∎

Consequently, a sufficient criterion for verifying the KLS conjecture is to establish that $P^{\infty}_{\sigma_{\partial\Omega}}\leq C^{\prime}$ for any isotropic convex $\Omega$ - a “weak KLS conjecture for cone measures”. This suggests that the most difficult part of the conjecture concerns the behavior of $1$-Lipschitz functions on the boundary.

It may be more desirable to work with the Lebesgue measure $\lambda_{\partial\Omega}$ instead of the cone measure $\sigma_{\partial\Omega}$. Since:

(see e.g. ), Theorem 2 together with (5) immediately yields:

For any smooth convex domain $\Omega$ with barycenter at the origin:

Note that for an isotropic convex body, the isoperimetric ratio term $I$ satisfies:

The left-hand side in fact holds for any arbitrary set $\Omega$ by the sharp isoperimetric inequality (11). The right-hand side follows since when $\Omega$ is convex and isotropic, it is known that $\Omega\supset\frac{1}{C}B_{2}^{n}$ (e.g. ). Consequently (see e.g. ):

and so (15) immediately follows. Up to the value of $C^{\prime}$, the right-hand side is also sharp, as witnessed by the $n$-dimensional cube. Note that by (13), $\left|\Omega\right|^{1/n}\simeq P^{Lin}_{\Omega^{*}}\leq P^{Lin}_{\Omega}=1$ for any isotropic convex body $\Omega$, and so in fact $I\leq C^{\prime\prime}\sqrt{n}$.

To avoid the isoperimetric ratio term $I$ which may be too large, we can instead invoke Theorem 5:

For any strictly convex smooth domain $\Omega$:

Note that by Jensen’s inequality and Remark 6:

but perhaps the term $A\frac{\left|\partial\Omega\right|}{\left|\Omega\right|}$ is nevertheless still more favorable than $I$.

For the proof, we require the following variant of the notion of $P^{\infty}_{\Omega}$:

It follows from the results of that for any convex $\Omega$:

Assuming that $f$ is $1$-Lipschitz and invoking Theorem 5, it follows that:

Applying this to $f-a$ where $a:=\int_{\partial\Omega}fd\lambda_{\partial\Omega}$, we obtain:

But $B=\frac{n}{n+1}A$ by Remark 6, and so the assertion follows from (17). ∎

2 A concrete bound

To control the variance of $1$-Lipschitz functions on the boundary $\partial\Omega$, we recall an argument from our previous work , where a generalization of the following inequality of A. Colesanti was obtained:

for any strictly convex $\Omega$ with smooth boundary and smooth function $f$ on $\partial\Omega$. Applying the Cauchy-Schwartz inequality, we obtain for any $1$-Lipschitz function $f$ with $\int_{\partial\Omega}fd\lambda_{\partial\Omega}=0$:

where $\kappa_{\partial\Omega}(x)$ denotes the (positive) minimal principle curvature of $\partial\Omega$ at $x$, so that $\text{II}_{\partial\Omega}\geq\kappa Id$. Consequently, the right-hand-side is an upper bound on $P^{1,\infty}_{\lambda_{\partial\Omega}}$. Using the equivalence (17) in a more general Riemannian setting, we were able to deduce in that:

Plugging this estimate into the estimates of the previous subsection, we obtain:

The easiest option is to invoke Corollary 9, but note that Corollary 8 or 10 would also work after an appropriate application of Cauchy-Schwartz. Coupled with (19), it follows that:

But since $P^{Lin}_{\Omega}\leq P^{N}_{\Omega}$, the assertion follows for e.g. $n\geq 8C_{1}$. ∎

Note that this estimate yields the correct result, up to constants, for the Euclidean ball. A concrete class of isotropic convex bodies for which the first term above $\frac{\left|\partial\Omega\right|}{\sqrt{n}}$ is upper bounded by a constant, is the class of quadratically uniform convex bodies $\Omega$, since in isotropic position $\Omega\supset c\sqrt{n}B_{2}^{n}$ and $\left|\Omega\right|^{1/n}\simeq 1$ (see e.g. ). It is not hard to show that when in addition $\Omega\subset C_{1}\sqrt{n}B_{2}^{n}$ - i.e. $\Omega$ is an isotropic quadratically uniform convex body which is isomorphic to a Euclidean ball - then $P^{N}_{\Omega}\leq C_{2}$. It would be very interesting to see if the additional assumption $\Omega\subset C_{1}\sqrt{n}B_{2}^{n}$ could be removed by employing the estimate given by Theorem 11.

3 Reduction to harmonic functions

We conclude this section by providing another different reduction of the KLS conjecture:

There exists a universal constant $C>1$ so that:

where $H$ denotes the class of harmonic functions $h$ on $\Omega$. In fact, for large enough $n$, one can use $C=2$.

Fix an arbitrary smooth function $f$ on $\Omega$, and solve the Poisson equation $\Delta h=0$, $h|_{\partial\Omega}=f|_{\partial\Omega}$. One has:

Since $f-h$ vanishes on $\partial\Omega$, the Faber-Krahn inequalities (2) or (10) imply:

But since $h$ is harmonic and $(f-h)|_{\partial\Omega}=0$ we have $\int\langle\nabla f-\nabla h,\nabla h\rangle d\lambda_{\Omega}=0$, and consequently:

It remains to note that since linear functions are harmonic, $P^{H}_{\Omega}\geq P^{Lin}_{\Omega}\geq P^{Lin}_{\Omega^{*}}\simeq\left|\Omega^{*}\right|^{2/n}$, concluding the proof. ∎

It is not clear to us if it enough to only control the variance of harmonic functions $h$, so that the restriction $h|_{\partial\Omega}$ is $1$-Lipschitz. The reason is that we do not know whether $h$ has bounded Lipschitz constant on the entire $\Omega$, and so we cannot apply (5). We believe that the latter would be an interesting property of convex domains which is worth investigating. A small observation in this direction is that $\left|\nabla h\right|^{2}$ is subharmonic and hence satisfies the maximum principle, but we do not know how to control the derivative in the normal direction to $\partial\Omega$.

Given a convex body $K$, consider the map $T(x)=\frac{x}{\left\|x\right\|_{K}}$ where $\left\|\cdot\right\|_{K}$ denotes the gauge function of $K$ (when $K$ is origin-symmetric, this function defines a norm). It is an elementary exercise to show that $T_{*}\mu=\sigma_{\partial K}$ if and only if $K=cK_{\mu}$ for some $c>0$ (see [27, Proposition 3.1]).

If $T(x)=\frac{x}{\left\|x\right\|_{K}}$ and $\partial K$ is smooth then:

where $h_{K}(\theta)=\sup\left\{\left\langle x,\theta\right\rangle;x\in K\right\}$ denotes the support function of $K$.

Since $\nabla\left\|x\right\|_{K}$ is parallel to $\nu=\nu_{\partial K}(T(x))$, taking the partial derivative in the direction of $x$ verifies that:

Consequently $dT(x)=\frac{1}{\left\|x\right\|}(Id-\frac{x\otimes\nu}{\left\langle x,\nu\right\rangle})$. Now observe that:

But $dT(x)dT(x)^{*}=\frac{1}{\left\|x\right\|^{2}}(Id+u\otimes u)$, where $u=\nu-\frac{x}{\left\langle x,\nu\right\rangle}$. Consequently, its top eigenvalue is:

It remains to note that when $x\in\partial K$ then $\left\langle x,\nu_{\partial K}(x)\right\rangle$ is precisely the support function of $K$ in the direction of the latter normal. Consequently, $\left\langle x,\nu\right\rangle=\left\|x\right\|_{K}h_{K}(\nu)$, and the assertion follows. ∎

It is easy to see that if the density $f$ of $\mu$ is smooth, then so is $\partial K_{\mu}$, and so by approximation we may assume that this is indeed the case. Consequently, if $K_{\mu}\supset RB_{2}^{n}$, we have by Lemma 15:

Integrating in polar coordinates, we have:

Denoting $k_{p}(\theta):=(p\int_{0}^{\infty}r^{p-1}f(r\theta)dr)^{1/p}$, we use that for any non-negative function $f$ on $[0,\infty)$:

It remains to apply a result of M. Fradelizi stating that for a log-concave measure $\mu=f(x)dx$ with barycenter at the origin:

Plugging all of these estimates into (20), the assertion is proved. ∎

In particular, if $\mu$ satisfies the KLS conjecture then so does $\lambda_{K_{\mu}}$, as soon as $\frac{\int\left|x\right|^{2}d\lambda_{K_{\mu}}}{R^{2}}$ is bounded above by a constant.

This result was already noticed by Bo’az Klartag and the second-named author using a more elaborate computation which was never published. The idea is to control the average Lipschitz constant of the radial map from pushing forward $\mu$ onto $\lambda_{K_{\mu}}$ instead of $\sigma_{\partial K_{\mu}}$.

We employ Corollary 8 and Proposition 14. When $n$ is large-enough, $C\frac{4}{n}P^{Lin}_{K_{\mu}}\leq\frac{1}{2}P^{Lin}_{K_{\mu}}\leq\frac{1}{2}P^{N}_{K_{\mu}}$, and hence by Corollary 8:

Denoting $T(x)=\frac{x}{\left\|x\right\|_{K_{\mu}}}$, we see by Proposition 14 that for any $1$-Lipschitz function $f$ on $\partial K_{\mu}$:

This implies the first part of the assertion.

The second part follows since, as shown by Ball (see for the non-even case):

We thus obtain a simple recipe for obtaining good spectral-gap estimates on certain convex bodies $K$ having in-radius $R$ so that $\int\left|x\right|^{2}d\lambda_{K}(x)/R^{2}$ is bounded above by a constant: if we can find a log-concave measure $\mu$ having good spectral-gap so that $K_{\mu}=K$, Theorem 16 will imply that $K$ also has good spectral-gap.

An inspection of the proofs of Proposition 14 and Theorem 16 shows that we may replaces $\frac{1}{R^{2}}$ in all of the occurrences above, with the more refined expression $\int_{\partial K_{\mu}}\frac{d\sigma_{\partial K_{\mu}}}{h^{2}_{K_{\mu}}(\nu_{\partial K_{\mu}})}$. However, we do not know how to effectively control the latter quantity.

Using Theorem 16, we avoid passing through the Schechtman–Zinn concentration results. Indeed, let $\mu_{p}$ denote the one-dimensional probability measure $\frac{1}{2\Gamma(1/p+1)}\exp(-\left|t\right|^{p})dt$. The $n$-fold product measure $\mu_{p}^{n}:=\mu_{p}^{\otimes n}$ has density $f_{p}^{n}(x)$ where:

Indeed, by inspecting the proof of Theorem 16 and employing Lemma 15, we see that we just need to control:

and the invariance under permutation of coordinates, we conclude that:

and so by a similar computation we conclude:

uniformly in $p\in[2,\infty]$, whereas it is elementary to verify that in that range:

Putting everything together, we see that:

It is natural to wonder whether the only inequality we have used to derive the above estimate, namely $\left|x\right|^{2}\leq n^{1-2/p}\left\|x\right\|_{p}^{2}$, was perhaps too crude. However, this is not the case, and unfortunately it is the method of working with the map $T(x)=x/\left\|x\right\|_{K_{\mu}}$ which is too crude. Indeed, when $p=\infty$, so that $\mu_{\infty}^{n}$ is the uniform measure on $^{n}$ and $K=K_{\mu_{\infty}^{n}}=[-1/2,1/2]^{n}$, we see by Lemma 15 that:

confirming that our estimate (23) is tight. This example suggests that perhaps it is better to work with the radial map from pushing forward $\mu$ onto $\lambda_{K_{\mu}}$ instead of our map $T$ which pushes $\mu$ onto $\sigma_{\partial K_{\mu}}$.

References