Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures

Introduction

Their conjecture was mainly motivated by the Central Limit Problem for log-concave measures, and as pointed out in , implies that most marginals of log-concave measures are approximately Gaussian.

Applied to the function $f(x)=|x|^{p}$ with $p=c\sqrt{n}$, the KLS conjecture implies (see and Section 4) that:

It was shown by G. Paouris that the predicted positive deviation estimate (1.2) indeed holds in the large:

Here $\left\|A\right\|_{op}$ denotes the operator norm of $A$. Recall that $X$ (and its density) is said to be “$\psi_{\alpha}$ with constant $b_{\alpha}$” if:

Note that this definition is linearly invariant and that necessarily $b_{\alpha}\geq 2^{-1/\alpha}$. We will simply say that “$X$ is $\psi_{\alpha}$”, if it is $\psi_{\alpha}$ with a universal positive constant $C$. By a result of Berwald or by Borell’s Lemma (see [32, Appendix III]), it is well known that any $X$ with log-concave density is $\psi_{1}$ with $b_{1}\leq C$, some universal constant, and so we only gain additional information when $\alpha>1$.

Subsequently, it was shown by Paouris that under the same assumptions, the following small-ball estimate, analogous to the large deviation one (1.4), also holds:

In a breakthrough work, the first non-trivial estimate on the concentration of $|X|$ around its expectation was given by B. Klartag in , involving delicate logarithmic improvements in $n$ over the trivial bounds. This validated the conjectured thin-shell concentration (1.1), allowing Klartag to resolve the Central Limit Problem for log-concave measures. A different proof continuing Paouris’ approach was given by Fleury, Guédon and Paouris in . Klartag then improved in his estimates from logarithmic to polynomial in $n$ as follows (for any small $\varepsilon>0$):

This implies in particular a thin-shell estimate of:

Note, however, that when $t=1/2$, (1.6) does not recover the sharp positive large-deviation estimate of Paouris (1.3).

Recently in , B. Fleury improved Klartag’s thin-shell estimate to:

by obtaining the following deviation estimates:

Note, however, that when $t=1/2$, Fleury’s positive and negative large-deviation estimates are both inferior to those of Klartag, and so in the mesoscopic scale $t=n^{-\delta}$ ($\delta>0$ small), Klartag’s estimates still outperform Fleury’s (and Paouris’ ones are inapplicable). In addition, note that both Klartag and Fleury’s estimates do not seem to improve under a $\psi_{\alpha}$ condition, contrary to the ones of Paouris. See also for further related results.

All of this suggests that one might hope for a concentration estimate which:

Recovers Paouris’ sharp positive large-deviation estimate (1.4).

Improves the best-known thin-shell estimate of Fleury.

Improves the best-known mesoscopic-deviation estimate of Klartag.

Interpolates continuously between all scales of $t$ (bulk, mesoscopic, large-deviation).

The aim of this work is to provide precisely such an estimate.

Following Paouris, we formulate our main results in greater generality, allowing an application of a linear transformation to $X$.

In particular, we obtain the following thin-shell estimate:

For concreteness and future reference, we state again the deviation estimates above and below the expectation separately: the constant $C$ in (1.7) may actually be removed in the former estimate:

and combining our estimate (1.7) with Paouris’ small-ball estimate (1.5), we obtain for the latter:

Applying Theorem 1.1 with $\alpha=1$ and $A=Id$, we obtain that for any isotropic $X$ with log-concave density, the above estimates hold with $\eta\geq cn$, and in particular we deduce the following improved thin-shell estimate:

Theorem 1.1 is a standard consequence of (and essentially equivalent to) the following moment estimates, which are the main result of this work:

With the same assumptions and notation as in Theorem 1.1, for any $1\leq\left|p-2\right|\leq c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}$:

and for any $c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}\leq\left|p-2\right|\leq c_{2}\eta^{\frac{\alpha}{2}}$:

This should be compared to the bound $D_{Che}(\mu)\geq c>0$ conjectured by Kannan, Lovász and Simonovits . Note that our estimate improves all the way to $D_{Che}(\mu)\geq cn^{-\frac{3}{8}}$ when the density of $\mu$ is $\psi_{2}$.

2 The Approach

For the proof of Theorem 1.2, we use many of the ingredients developed previously by Klartag , and adapted to the language of moments by Fleury :

It is (almost) enough to verify (1.13) and (1.14) with $AX$ replaced by $Y=(AX+G_{n})/\sqrt{2}$.

It is useful to first project $Y$ onto a lower-dimensional subspace $F\in G_{n,k}$. This idea also appears in essence in the work of Paouris . Klartag and Paouris use V. Milman’s approach to Dvoretzky’s theorem for identifying lower-dimensional structures in most marginals $P_{F}Y$. Fleury, on the other hand, takes an average over the Haar measure on $G_{n,k}$, which is more efficient (see or below):

Rewriting using the invariance of the Haar measure and polar coordinates:

To control the ratio in (1.16), a good bound on the log-Lipschitz constant $L_{k,p}$ of $h_{k,p}$ is required.

Our main technical result in this work is the following improvement over the log-Lipschitz bounds of Klartag from :

Under the same assumptions as in Theorem 1.1, if $p\geq-k+1$ then $L_{k,p}\leq C\left\|A\right\|_{op}b_{\alpha}\max(k,p)^{1/\alpha+1/2}$.

Contrary to Klartag’s analytical approach for controlling the log-Lipschitz constant, ours is completely based on geometric convexity arguments, employing the convex bodies $K_{k+q}$ introduced by K. Ball in , and a variation on the $L_{q}$-centroid bodies, which were introduced by E. Lutwak and G. Zhang in .

Fleury proceeds by employing three additional ingredients:

Consequently, $p\mapsto\log(h_{k,p}(u)/\Gamma(k+p))$ is concave on $p\in[-k+1,\infty)$ for any fixed $u\in SO(n)$. This ingredient was also used in .

We proceed by using these ingredients as our predecessors, but our proof corrects the slight inefficiency of Fleury’s approach in the resulting large-deviation estimate (witnessed by the comparison to Klartag’s estimate earlier). The improvement here comes from the fact that we take the derivative in $p$ of (1.15), and optimize on the dimension $k$ for each $p$ separately, as opposed to optimizing on a single $k$ directly in (1.15). However, this by itself would not yield the improvement in the thin-shell estimate - the latter is due to our improved log-Lipschitz estimate in Theorem 1.5. Only by combining this improved log-Lipschitz estimate with our variation on Fleury’s method, are we able to recover the sharp large-deviation estimates of Paouris (1.4). Moreover, the negative moment estimates of (1.13) and (1.14) are also obtained almost for free, at least with $AX$ replaced by $Y$, after some slight additional justification for handling the $p$ moments in the range $p\in[-c\eta^{-1/2},c\eta^{-1/2}]$.

The rest of this work is organized as follows. In Section 2 we prove a more general version of Theorem 1.5. In Section 3 we provide a complete proof of a refined version of Theorem 1.2, with $AX$ replaced by $Y$, without eluding to (1.5). In Section 4, we derive for completeness Theorem 1.1 from Theorem 1.2, and obtain the reduction from $AX$ to $Y$. In the Appendix, we provide a proof of Proposition 2.6 and other lemmas, whose purpose is to handle the case when $X$ is not centrally-symmetric (has non-even density).

Acknowledgement. We thank Bo’az Klartag for his interest and comments and Matthieu Fradelizi for discussions. We also thank the anonymous referees for helpful suggestions. This work was done in part when the authors attended the Thematic Program on Asymptotic Geometric Analysis at the Fields Institute in Toronto.

An improved log-Lipschitz estimate

Note that $\pi_{E}g$ is log-concave for any $E\in G_{n,k}$ by the Prékopa–Leindler Theorem (e.g. ).

was obtained by Klartag [24, Lemma 3.1], playing a crucial role in his polynomial estimates on the thin-shell of an isotropic log-concave measure. When $t\leq C\sqrt{k}$, Klartag’s estimate is of the order of $k^{2}$. In , Fleury defined a truncated version of (2.1), where the integral ranges up to $C\sqrt{k}$. Klartag’s estimate obviously implies the same bound on the log-Lipschitz constant of this truncated version of $h_{k,p}$.

Our main technical result in this work is the following improved estimate on the log-Lipschitz constant of $h_{k,p}$, which is completely based on geometric convexity arguments. Note that we do not need any truncation, nor do we need to assume that $Y$ has been convolved with a Gaussian to obtain a meaningful estimate. However, the improvement over Klartag’s $k^{2}$ bound appears after this convolution.

(here as usual $a_{+}:=\max(a,0)$). A dual variant of this definition (when $q\in(0,1)$) was also used by C. Haberl in . When $w$ is even, this coincides with the more standard definition of the $L_{q}$-centroid body, introduced by E. Lutwak and G. Zhang in (under a different normalization):

A very useful result for handling the non-even case is due to Grünbaum (see also [17, Formula (10)] or [7, Lemma 3.3] for simplified proofs):

Note that by definition, $Y$ (and its density $g$) is $\psi_{\alpha}$ ($\alpha\in$) with constant $b_{\alpha}$ iff $Z_{q}(g)\subset b_{\alpha}q^{1/\alpha}Z_{2}(g)$ for all $q\geq 2$. Also recall that by a result of Berwald or as a consequence of Borell’s Lemma (see also or [32, Appendix III]), a log-concave probability density $g$ is always $\psi_{1}$, and that moreover:

If in addition the barycenter of $g$ is at the origin, then repeating the argument leading to (2.2) and using Lemma 2.2, one verifies:

When $g$ is isotropic, note that $Z_{2}(g)=B_{2}^{n}$, and one may similarly show (see Lemma A.4) that $cB_{2}^{n}\subset Z^{+}_{2}(g)\subset\sqrt{2}B_{2}^{n}$. It follows immediately from (2.3) that in that case $\textrm{dist}(Z^{+}_{k}(g),B_{2}^{n})\leq Ck$, and we see that Theorem 2.1 recovers Klartag’s $k^{2}$ order of magnitude when $p\leq k$ (which is the case of interest in the subsequent analysis).

The improvement over Klartag’s bound comes from the following elementary:

$Z_{q}^{+}(g)\supset c\sqrt{q}B_{2}^{n}$.

If $X$ is $\psi_{\alpha}$ ($\alpha\in$) with constant $b_{\alpha}$, then $Z_{q}^{+}(g)\subset(C_{1}\left\|A\right\|_{op}b_{\alpha}q^{1/\alpha}+C_{2}\sqrt{q})B_{2}^{n}$.

Given $\theta\in S^{n-1}$, denote $Y_{1}=\pi_{\theta}Y$, $X_{1}=\pi_{\theta}AX$ and $G_{1}=\pi_{\theta}G_{n}$ (a one-dimensional standard Gaussian random variable). We have:

Assuming that $X$ is $\psi_{\alpha}$ with constant $b_{\alpha}$ and isotropic, it follows that:

and the second assertion readily follows. ∎

Consequently, when $\left\|A\right\|_{op}\geq 1$, Theorem 2.1 implies that:

2 Proof of Theorem 2.1

To this end, we recall the following crucial fact, due to K. Ball [3, Theorem 5] in the even case, and verified to still hold in the general one by Klartag [22, Theorem 2.2]:

Finally, note that since $B_{2}^{m}$ is centrally-symmetric, then $C_{1}B_{2}^{m}\subset K\cap-K\subset K\subset C_{2}B_{2}^{m}$ iff $C_{1}B_{2}^{m}\subset K\subset C_{2}B_{2}^{m}$, and hence:

where $c_{p,k}=\textrm{Vol}(S^{k-1})/(k+p)$ is totally immaterial. Consequently:

Since $\left|\frac{d}{ds}\left\|u_{s}(\theta_{0})\right\|_{K_{k+p}(\pi_{E_{0}}g)}\right|\leq\left\|\frac{d}{ds}u_{s}(\theta_{0})\right\|_{\hat{K}_{k+p}(\pi_{E_{0}}g)}$ by the triangle-inequality (2.5), we conclude using (2.6) that:

2.2 Type-2 movement

Performing the change of variables $r=vt$, which is valid except at the negligible point $t=0$, we obtain:

where $c_{p,k}=\textrm{Vol}(S^{k-1})/(k+p+1)$. Using that $\frac{d}{ds}\xi_{s}=-\theta_{s}$ and the triangle inequality (2.5) and (2.6) for $\left\|\cdot\right\|_{K_{k+p+1}(\pi_{H}g)}$, we obtain:

where we have used the fact that $\theta_{0}$ and $\xi_{0}$ are orthogonal unit vectors in the last equality.

2.3 Type-3 movement

Finally, we analyze the most important movement type, which is responsible for a subspace of movements of dimension $(k-1)(n-k)$ (out of the $\text{dim }G_{n,k}+\text{dim }S^{k-1}=k(n-k)+(k-1)$ dimensional subspace of non-trivial movements).

Let $0\neq B\in T_{3}$ generate a Type-3 movement $\left\{u_{s}\right\}$, and set $e^{j}_{s}:=u_{s}(e^{j})$ and $f^{j}:=\frac{d}{ds}e^{j}_{s}|_{s=0}$, $j=2,\ldots,k$. The Type-3 movement ensures that $u_{s}(\theta_{0})=\theta_{0}$ and that all $f^{j}\in E_{0}^{\perp}$. Denote $F_{0}:=\text{span}\{f^{2},\ldots,f^{k}\}$, and note that by slightly perturbing $B$ if necessary, we may assume that $F_{0}$ is $k-1$ dimensional. Finally, set $H=E_{0}\oplus F_{0}\in G_{n,2k-1}$, and notice that $H$ is invariant under $u_{s}$ (since $u_{s}$ is an isometry acting as the identity on the orthogonal complement). Consequently, $H=E_{s}\oplus F_{s}$, where $E_{s}:=u_{s}(E_{0})$ and $F_{s}:=u_{s}(F_{0})$, and therefore:

Using the change of variables $y=zt$, we obtain (with $c_{p,k}=\textrm{Vol}(S^{k-1})/(2k-1+p)$):

which we rewrite, since $u_{s}$ is orthogonal, as:

As usual, the triangle inequality (2.5) for $\left\|\cdot\right\|_{K_{2k-1+p}(\pi_{H}g)}$ implies that:

where we have used that $\theta_{0}$ is perpendicular to $F_{0}$, and that $\left\|B\right\|_{op}\leq\left\|B\right\|_{HS}/\sqrt{2}$ for any anti-symmetric matrix $B$, as may be easily verified by using the Cauchy–Schwarz inequality.

𝑚𝑝K_{m+p} to Euclidean ball To conclude the proof of Theorem 2.1, it remains to control the geometric distance of $K_{m+p}(\pi_{H}g)$ to a Euclidean ball, for $H\in G_{n,m}$ with $m$ of the order of $k$. To this end, we compare $K_{m+p}(\pi_{H}g)$ to $Z_{q}(\pi_{H}g)=P_{H}Z_{q}(g)$ for a suitably chosen $q\geq 1$. Our motivation comes from the groundbreaking work of Paouris , who noted that:

and using the inclusion $Z_{q}(K)\subset conv(K\cup-K)$ for any set $K$ of volume $1$, obtained an upper bound on $\textrm{Vol}(Z_{q}(\pi_{H}g))$ by bounding above $\textrm{Vol}(K_{m+q}(\pi_{H}g))$, enabling Paouris to deduce important features of $P_{H}Z_{q}(g)$. In this work, on the other hand, we take the converse path, passing from $K_{m+q}$ bodies to $Z_{q}$ ones, and consequently need to introduce the $Z^{+}_{q}$ bodies to handle non-even densities. Moreover, we require bounds on $Z^{+}_{q}(K)$ both from above and from below, which turn out to be more laborious in the non-even case (when $K$ is not centrally-symmetric).

Since the distance to the Euclidean ball cannot increase under orthogonal projections, and since $c_{1}Z^{+}_{k}(g)\subset c_{2}Z^{+}_{m}(g)\subset c_{3}Z^{+}_{2k-1}(g)\subset c_{4}Z^{+}_{k}(g)$ when $k\leq m\leq 2k-1$ by (2.3), it remains to establish the following:

For the proof, we recall several useful properties of the bodies $K_{q}(w)$ and $Z^{+}_{q}(K)$. First, it is known (see for the even case and [22, Lemmas 2.5,2.6] or [35, Lemma 3.2 and (3.12)] for the general one) that under the assumptions of Theorem 2.5:

Second, integration in polar coordinates (cf. ) directly shows that:

Lastly, we require the following proposition, which is well-known in the even-case (e.g. [33, Lemma 4.1]), but requires more work in the general one (note for instance that the barycenter of $K_{m+q}(w)$ below need not be at the origin); its proof is postponed to the Appendix.

When $p\geq 1$, observe that (2.9), (2.8) and Stirling’s formula, imply that:

and so when $p\geq m$ the asserted claim follows. Otherwise, using (2.7), Stirling’s formula, (2.9) and (2.8), we see that if $q\geq\max(p,1)$ then:

Setting $q=m$, the case $p<m$ is also settled. ∎

The proof of Theorem 2.1 is now complete.

Moment Estimates

Note that by the Prékopa–Leindler Theorem, $Y$ itself has log-concave density. We also remark that it is possible to improve the moment estimates in the range $1\leq\left|p-2\right|\leq c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}$ exactly as in Theorem 1.2, but we do not insist on this here.

Passing to polar coordinates on $F\in G_{n,k}$ and using the invariance of the Haar measures on $G_{n,k}$, $S(F)$ and $SO(n)$ under the action of $SO(n)$, we verify that:

where $U$ is uniformly distributed on $SO(n)$.

2 Controlling the derivative

We now deviate from Fleury’s argument and proceed to estimate:

A useful fact, easily verified by direct calculation, is that:

We proceed with estimating (3.4). As explained:

Our main idea here is to decompose the numerator as follows:

The contribution of the second term in (3.6) is controlled using the log-Sobolev inequality (1.19):

where recall $L_{k,p}$ denotes the log-Lipschitz constant of $u\mapsto h_{k,p}(u)$. To control the contribution of the first term in (3.6), we first write given $u\in SO(n)$:

By Borell’s concavity result (1.18), we realize that:

Plugging this estimate back into (3.5) and (3.6), we obtain:

By using the Jensen and Cauchy–Schwarz inequalities, we bound the second term by:

Now, plugging all the estimates (3.7), (3.8), (3.9) into (3.5) using the decomposition (3.6), and plugging the result into (3.4), we obtain:

3 Optimizing on the dimension

As observed by Fleury in , using that the function $\frac{d}{dp}\log\Gamma(p)$ is concave, one easily verifies that the last term above satisfies:

Since the contribution of this term is insignificant relative to the second one, we simply use (3.10) as an upper bound. For the second term, for any $q\neq 0$ having the same sign as $p$ and such that $k+p+q>0$, we estimate using Jensen’s inequality:

Applying Stirling’s formula, setting $q=(p+k-1)\frac{p}{k-1}$, which indeed satisfies the above restrictions since $p\geq-\frac{k-1}{2}$, and using the latter condition on $p$, one verifies that:

see also Remark 3.3 below for an alternative derivation. Plugging our estimates for $L_{k,q}$ obtained in Corollary 2.4, and noting that $\left\|A\right\|_{op}\geq 1$ since $\left\|A\right\|_{HS}^{2}=n$, we conclude that if $X$ is $\psi_{\alpha}$ ($\alpha\in$) with constant $b_{\alpha}$, then:

for all integers $k$ in $[\max(2,2\left|p\right|+1),n]$. Optimizing on $k$ in that range, we set:

which is guaranteed to be in the desired range whenever $\left|p\right|\in[4\eta^{-1/2},\frac{1}{64}\eta^{\alpha/2}]$, as may be easily verified using that $\left\|A\right\|_{op}\geq 1$ and $b_{\alpha}\geq 2^{-1/\alpha}$. Consequently, for such $p$, we obtain:

Setting $p_{0}:=4\eta^{-1/2}$, we may assume that $p_{0}\leq 2$ since $\eta$ was assumed in the Introduction to be large enough (otherwise the statement of Theorem 3.1 follows easily), and so integrating over $p$ and adjusting constants, we obtain:

4 Moments near 00

It remains to bridge the gap between the $p_{0}$ and $-p_{0}$ moments. Note that since we assume that $p_{0}\leq 2$ and that $n$ is larger than some constant, then $p_{0}\leq\frac{k_{0}-1}{2}$ for e.g. $k_{0}=5$. Unfortunately, in the range $p\in[-p_{0},p_{0}]$, our key estimate (3.12) only yields (using $k=k_{0}$):

which in particular is not integrable at . We consequently treat this gap differently, by reproducing Fleury’s argument from .

Note that by Borell’s concavity result (1.18), we have:

Using the reverse Hölder inequality (1.20) for comparing the $L_{1/2}$ and $L_{1}$ norms of $h_{k_{0},p_{0}}$ and $h_{k_{0},-p_{0}}$, we obtain:

By Corollary 2.4 we know that $L_{k_{0},p_{0}},L_{k_{0},-p_{0}}\leq C_{3}\left\|A\right\|_{op}b_{\alpha}k_{0}^{1/\alpha+1/2}$, and we conclude that:

Finally, using (3.2), (3.3) and (3.10), we see that:

This fills the remaining gap, and together with (3.13) and (3.14), the assertion of Theorem 3.1 follows.

Deviation Estimates

A completely standard consequence of Theorem 3.1 is the following:

With the same assumptions and notation as in Theorem 3.1:

and note that there exists a constant $t_{0}\in(0,1]$, so that:

Here $c,C>0$ are the two constants appearing in Theorem 3.1, which guarantee that:

Since $\frac{1+t}{1+t/2}\geq 1+t/3$ for $t\in$, we obtain by the Markov–Chebyshev inequality:

Expressing $p_{1}$ as a function of $t$ for $t$ in the range specified in (4.4), and plugging this above, we obtain:

To extend this estimate to the entire interval $[0,t_{0}]$, note that:

and so adjusting the constants appearing above:

Finally, a standard application of Borell’s lemma (e.g. as in ), ensures that:

concluding the proof of the positive deviation estimate (4.1).

Expressing $p_{2}$ as a function of $t$ for $t$ in the range specified in (4.4), and plugging this above, we obtain:

Adjusting the value of $C_{2}$ above, the estimate extends to the entire range $t\in[0,t_{0}]$. Lastly, setting $p_{3}=-c_{3}\eta^{\frac{\alpha}{2}}$ so that:

we obtain for all $\varepsilon\in(0,1/2)$:

Adjusting all constants, the negative deviation estimate (4.2) follows. ∎

To conclude the proof of Theorems 1.1 and 1.2, we estimate the deviation of $AX$ by that of $Y$ exactly like Klartag . Indeed, according to the argument described in the proof of [23, Proposition 4.1], we have:

for some universal constant $C>1$. The deviation estimate (1.7) of Theorem 1.1 immediately follows from the corresponding estimates of Theorem 4.1. However, the more refined deviation estimates (1.10) and (1.11) do not follow: (1.10) only follows up to the unnecessary constant $C$ in front of the estimate:

and (1.11) follows without the decay to as $t\rightarrow 1$:

To resolve these last issues, we proceed as follows. The unnecessary constant $C>1$ in (4.5) is easily removed e.g. by repeating the argument of Fleury from . Indeed, when $p\geq 1$, by the symmetry and independence of $G_{n}$, convexity of $t\mapsto t^{p}$ and the Cauchy–Schwarz inequality, we have:

Consequently, the $p$-moment estimates of Theorem 3.1 hold equally true (after adjusting constants) with $Y$ replaced by $AX$, when $p\geq 3$. In particular, the $p$-moment estimates (1.14) of Theorem 1.2 for $p\geq c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}$ are obtained. Repeating the relevant parts in the proof of Theorem 4.1, the desired positive deviation estimate (1.10) follows. Finally, applying [14, Lemma 6] to the deviation estimates of Theorem 1.1, the positive $p$-moment estimates are improved in the range $1\leq p-2\leq c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}$, obtaining the right-hand side of (1.13); see also below for a sketch of an alternative derivation. This takes care of the positive moment and deviation estimates.

Reducing from $AX$ to $Y$ the small-ball estimate (1.11), or equivalently, the negative moment estimates of (1.14), seems more involved, and further arguments are needed. We choose to bypass these here by simply employing Paouris’ small-ball estimate (1.5), which together with (4.6) yields for some $c_{3}\leq 1$ the desired:

The negative moment estimates of (1.13) and (1.14) then follow by integrating (4.8) by parts. Since the computation is not entirely straightforward when $1\leq\left|p-2\right|\leq c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}$, we sketch the argument, which is based on Fleury’s derivation in [14, Lemma 6] of positive moment estimates from deviation estimates. However, Fleury’s technique does not seem to generalize to negative moments, and so we provide an alternative proof, which is equally applicable to both positive and negative moments.

Assuming for simplicity that $p\geq 2$, we use (4.8) to bound the first integral above, evaluating separately the intervals $[1-c_{3}^{2}/2,1]$, $[0,1/p]$ and $[1/p,1-c_{3}^{2}/2]$, and (1.10) to bound the second integral, evaluating separately the intervals $[0,1/p]$ and $[1/p,\infty)$. Using the obvious estimates:

When $2\leq p\leq c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}$, this implies using $(1+2px)^{\frac{1}{2p}}\leq 1+x$:

In this range of values for $p$, $1/p\geq c(p/\eta^{\frac{\alpha}{2}})^{\frac{1}{1+\alpha}}$, and hence the integrand in the term involving $C_{8}$ is monotone decreasing. A standard computation then confirms that, in this range, both integrals involving $C_{8}$ and $C_{9}$ are dominated by the one involving $C_{7}$, yielding the negative moment estimates of (1.13); a similar argument does the job in the positive moment range. When $c_{1}\eta^{\frac{\alpha}{2(\alpha+2)}}\leq p\leq c_{2}\eta^{\alpha/2}$, we similarly verify from (4) that:

Bounding the second (dominant) term using the Laplace method, we obtain the negative moment estimates of (1.14), thereby concluding the proof of Theorem 1.2.

Appendix

In the Appendix, we prove several properties of the bodies $Z_{q}^{+}(K)$ (for $q\geq 1$) which are needed for the results of Section 2.

The right inequality is straightforward from the definitions. The left inequality is derived by following the proof of [33, Lemma 4.1], which uses the fact that the $1/(m-1)$ power of any one-dimensional marginal of $K$ is a concave function. ∎

To control the left-most term in Lemma A.1, we have:

This is essentially folklore (see e.g. [17, Lemma 1.1]), but we include a proof for completeness. We refer the interested reader e.g. to for the study of functional inequalities in the case of non-symmetric log-concave measures.

Plugging this into (A.1) and using Stirling’s formula, we verify that:

Now decomposing $\textrm{Vol}=\textrm{Vol}|_{H_{\theta}^{+}}+\textrm{Vol}|_{H_{-\theta}^{+}}$, (A.2) and (A.3) imply the assertion. ∎

Applying Lemma A.1 with $K=K_{m+q}(w)$ and using Lemma A.3, we obtain for all $\theta\in S^{m-1}$:

Lemma A.2 together with Lemma A.3 imply that:

Rearranging terms, the assertion of Proposition 2.6 follows. ∎

Applying now the reverse comparison using the right-hand side of (2.7) for both directions $\theta$ and $-\theta$, and summing the resulting estimates, we obtain:

Since the barycenter of $g_{0}$ is at the origin, we know by Lemma 2.2 that:

Together with (A.4), the assertion follows with e.g. $c=(3e^{2}(1+(e-1)^{3}))^{-1/2}$. ∎

References