arXiv subjects

Boaz Klartag

Publications and source records attributed to Boaz Klartag.

9 recordsLinked to original sources

arXiv2026

Digesting the proof of the sharp thin-shell inequality

We present a proof that determines the optimal value of the universal constant in the thin-shell theorem for log-concave distributions in high dimensions. We prove that for any log-concave random vector $X = (X_1,\ldots,X_n)$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ {\rm Var}( |X|^2 ) \leq 8 n. $$ The constant $8$ is optimal: equality is attained when $X_1,\ldots,X_n$ are independent, identically distributed, standard, centered exponential random variables. Moreover, among isotropic random vectors distributed uniformly on convex bodies in $\mathbb{R}^n$, the quantity ${\rm Var}(|X|^2)$ is maximized by the uniform distribution on a regular simplex. We also provide a corresponding sharp bound on the Hilbert-Schmidt norm of the tensor of $3^{rd}$-moments of isotropic, log-concave distributions. The argument relies on the analysis of a weighted Riemannian manifold associated with log-concave moment measures and the Monge-Ampère equation. This manifold was studied in this context in \cite{lc_moment}. The main improvement over \cite{lc_moment} comes from a concise yet effective analysis of the $3^{rd}$-derivatives tensor of the potential. The proof was found by GPT-5.6 Pro in response to prompts supplied by the first-named author, following general discussions between the two authors concerning log-concave moment measures. The prompts referred to the paper ``Logarithmically-concave moment measures I'' and suggested bootstrapping a bound on the second trace moment.

math.MG↗

arXiv2026

Distances between non-symmetric convex bodies: optimal bounds up to polylog

In this paper we determine, up to polylogarithmic factors, the diameter of the Banach--Mazur compactum of $n$-dimensional convex bodies without symmetry assumptions. We prove that for any convex bodies $K_1,K_2\subset \mathbb{R}^n$, \begin{equation} d_{BM}(K_1,K_2)\le Cn\log^α(n+1), \label{eq_1624} \end{equation} for universal constants $C,α>0$, improving an earlier bound of Rudelson. We also study the partial-containment distance $d_{PC}$, in which the Banach-Mazur requirement to contain the other body in its entirety is relaxed to $99\%$-containment. We prove that this relaxation leads to a very different behavior: \begin{equation} d_{PC}(K_1,K_2) \leq C \log^α (n+1) \label{eq_1625} \end{equation} for all convex bodies $K_1,K_2 \subseteq \mathbb{R}^n$. This demonstrates that in high dimensions, any convex body is not too far from an affine image of any other convex body, when we look at the bulk of their mass. \medskip In the centrally-symmetric case, the optimal upper bound for the Banach-Mazur distance is obtained in the John position. In contrast, our proofs rely on the isotropic position. The analytic core of our argument is a two-sided comparison, in the gauge order, between isotropic log-concave measures and Gaussian measures. This yields a new isotropic $M$-bound that complements E. Milman's $M^*$-bound. We also provide applications to linear symplectic geometry and to the first Dirichlet eigenvalue of the Laplacian.

math.MG↗

arXiv2026

Poisson approximation of random lattices

Fix a subset $S \subset \mathbb{R}^n$ of volume at most $c n$ that satisfies $S \cap (-S) = \emptyset$. We consider two point processes in $S$: the first is the Poisson point process of intensity one, and the second is the restriction of a random lattice to $S$, where the random lattice is distributed uniformly in the space of covolume-one lattices. We show that the total variation distance between these two point processes is at most $C e^{-c' n}$, where $c, C, c' > 0$ are universal constants.

math.PR↗

arXiv2026

Thin-shell bounds via parallel coupling

We prove that for any log-concave random vector $X$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ \mathbb{E} (|X| - \sqrt{n})^2 \leq C $$ where $C > 0$ is a universal constant. Thus, most of the mass of the random vector $X$ is concentrated in a thin spherical shell, whose width is only $C / \sqrt{n}$ times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of $X$ related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture.

math.PR↗

arXiv2026

Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid

We prove that in any dimension $n$ there exists an origin-symmetric ellipsoid ${\mathcal{E}} \subset {\mathbb{R}}^n$ of volume $ c n^2 $ that contains no points of ${\mathbb{Z}}^n$ other than the origin, where $c > 0$ is a universal constant. Equivalently, there exists a lattice sphere packing in ${\mathbb{R}}^n$ whose density is at least $cn^2 \cdot 2^{-n}$. Previously known constructions of sphere packings in ${\mathbb{R}}^n$ yielded densities of at most $C n \log n \cdot 2^{-n}$. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least $c n^2$ lattice points on its boundary, while containing no lattice points in its interior except for the origin.

math.MG↗

arXiv2025

Entropy and Learning of Lipschitz Functions under Log-Concave Measures

We study regression of $1$-Lipschitz functions under a log-concave measure $μ$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $μ$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $μ$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(μ)$. When this rate matches the Gaussian one, we show that both estimators achieve minimax bounds over a wide range of parameters. A key ingredient is sharp entropy estimates for the class of $1$-Lipschitz functions in $L^2(μ)$, which are new even in the Gaussian setting.

math.PR↗

arXiv2025

Polynomial Approximation in $ L^2 $ of the Double Exponential via Complex Analysis

We study the polynomial approximation problem in $L^2(μ_1)$ where $μ_1(dx) = e^{-|x|}/2 dx$. We show that for any absolutely continuous function $f$, $$ \sum_{k=1}^{\infty} \log^2(e+k) \langle f, P_k \rangle^2 \ \leq C \left( \int_{\mathbb{R}} \log^2(e+\lvert x \rvert) f^2 \, dμ_1 \ + \ \int_{\mathbb{R}} (f')^2 \, dμ_1 \right) $$ for some universal constant $C>0$, where $(P_k)_{k \in N}$ are the orthonormal polynomials associated with $μ_1$. This inequality is tight in the sense that $\log^2(e +k)$ on the left hand-side cannot be replaced by $a_k \log^2(e +k)$ with a sequence $a_k \longrightarrow \infty$. When the right hand-side is bounded this inequality implies a logarithmic rate of approximation for $f$, which was previously obtained by Lubinsky. We also obtain some rates of approximation for the product measure $μ_1^{\otimes d}$ in $\mathbb{R}^d$ via a tensorization argument. Our proof relies on an explicit formula for the generating function of orthonormal polynomials associated with the weight $\frac{1}{2\cosh(πx/2)}$ and some complex analysis.

math.CA↗

arXiv2024

Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound

We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq \mathbb{R}^n$ such that $$ Vol_{n-1}(K \cap H) > c, $$ where $c > 0$ is a universal constant. Our proof combines Milman's theory of $M$-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.

math.MG↗

arXiv2006

On volume distribution in 2-convex bodies

We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, we improve for 1<p<=2 some bounds on the isotropic constant of quotients of subspaces of L_p and S_p^m, the Schatten Class space.

math.FA↗