arXiv subjects

Natalia Tziotziou

Publications and source records attributed to Natalia Tziotziou.

8 recordsLinked to original sources

arXiv2026

Geometry of the subgaussian body of an isotropic convex body

For a centered convex body $K\subset\mathbb{R}^n$, let $Ψ_2(K)$ denote the symmetric convex body whose support function is given by the $ψ_2$-norms of linear functionals on $K$. The recent solution of Milman's problem on the existence of subgaussian directions by Letwin and Mikulincer naturally motivates the study of the geometry of this body. We prove that $Ψ_2(K)$ has bounded volume ratio with respect to the $L_2$-centroid body $Z_2(K)$. In the isotropic case, we also obtain sharp estimates for its mean width and the volume radii of its orthogonal projections, and derive consequences for the existence of subgaussian orthonormal bases. In particular, we construct orthonormal bases with quantitatively controlled subgaussian constants for every isotropic convex body.

math.MG↗

arXiv2026

Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures

This paper is dedicated to two geometric problems associated to log-concave measures on $\mathbb{R}^n$. First, we study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $μ$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. We prove that for every pair of symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $λ\in(0,1)$, $$μ(λK+(1-λ)L)^{c_n} \geq λμ(K)^{c_n}+(1-λ)μ(L)^{c_n},$$ where $c_n\geq c/n^3\ln n$ for some absolute constant $c>0$. Secondly, we study the maximal perimeter $Γ(μ)$ of an isotropic log-concave measure $μ$, without symmetry assumptions. We prove that $$Γ_n = \sup\{Γ(μ): \ μ\ \mbox{is an isotropic log-concave measure on } \mathbb{R}^n \} \approx n.$$ A key ingredient in both our proofs is a bound due to Eldan and Klartag (2008), which states that $$\int_{\mathbb{R}^n} |\nablaψ|\,dμ\leq Cn$$ for every isotropic log-concave probability measure $μ$ on $\mathbb{R}^n$ with density $e^{-ψ}$. We also present further applications of this estimate to projections of log-concave functions projections, moment and surface area measures of isotropic log-concave functions, highlighting the central role of the gradient of the logarithmic potential in high-dimensional convexity.

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arXiv2026

Moments of the Cramér transform of log-concave probability measures

Let $μ$ be a centered log-concave probability measure on ${\mathbb R}^n$ and let $Λ_μ^{\ast}$ denote the Cramér transform of $μ$, i.e. $Λ_μ^{\ast}(x)=\sup\{\langle x,ξ\rangle-Λ_μ(ξ):ξ\in\mathbb{R}^n\}$ where $Λ_μ$ is the logarithmic Laplace transform of $μ$. We show that $\mathbb{E}_μ\left[\exp\left(\frac{c_1}{n}Λ_μ^{\ast }\right)\right]<\infty $ where $c_1>0$ is an absolute constant. In, particular, $Λ_μ^{\ast}$ has finite moments of all orders. The proof, which is based on the comparison of certain families of convex bodies associated with $μ$, implies that $\|Λ_μ^{\ast}\|_{L^2(μ)}\leqslant c_2n\ln n$. The example of the uniform measure on the Euclidean ball shows that this estimate is optimal with respect to $n$ as the dimension $n$ grows to infinity.

math.MG↗

arXiv2026

Inequalities for sections and projections of log-concave functions

We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function $f$ and obtain upper and lower estimates for them in terms of the integral $\|f\|_1$ of $f$, we give estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problem, and we extend to log-concave functions the affirmative answer to a variant of the Busemann-Petty and Shephard problems, proposed by V. Milman. The main goal of this article is to show that the assumption of log-concavity leads to inequalities in which the constants are of the same order as that of the constants in the original corresponding geometric inequalities.

math.MG↗

arXiv2026

Regular functional covering numbers

We establish the existence of a regular functional $M$-position, in the sense of Pisier, for geometric log-concave functions. This provides a functional analogue of Pisier's regular $M$-positions for convex bodies and yields uniform control of covering numbers at all scales. Specifically, we show that every isotropic geometric log-concave function $f:\mathbb{R}^n \to [0,\infty)$ satisfies, for all $t\geq 1$, $$\max \left\{N(f, t \cdot g),\,N(f^*, t \cdot g),\,N(g, t \cdot f),\,N(g, t \cdot f^*)\right\} \leq \exp\left( \frac{γ_n^2\, n}{t} \right),$$ where $f^*$ denotes the Legendre dual of $f$, $(t \cdot f)(x)=f(x/t)$ is the $t$-homothety of $f$, $g(x)=\exp \left(-\frac{1}{2}|x|^{2}\right)$ and $γ_n \leq c(\ln n)^2$. Our result shows that the isotropic position of a log-concave function already provides an almost $1$-regular functional $M$-position.

math.MG↗

arXiv2026

On the maximal perimeter of isotropic log-concave probability measures

We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}^n$. For a measure $μ$, this quantity, denoted by $Γ(μ)$, is defined as the supremum of the $μ$-perimeter over all convex bodies and measures the largest possible boundary contribution of convex sets with respect to $μ$. Let $$Γ_n := \sup\{Γ(μ) : μ\text{ is an isotropic log-concave probability measure on } \mathbb{R}^n\}.$$ We prove that $Γ_n \leqslant Cn^{3/2}$, where $C>0$ is an absolute constant. This result improves the previously known $O(n^2)$ upper bound. Under additional structural assumptions, we obtain sharp linear bounds of order $O(n)$.

math.MG↗

arXiv2025

Sections and projections of the outer and inner regularizations of a convex body

We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santaló point is at the origin, with those of its inner and outer regularizations. We also provide functional extensions of these inequalities to the setting of log-concave functions. Our approach relies on the recent optimal $M$-estimate of Bizeul and Klartag for isotropic convex bodies.

math.MG↗

arXiv2025

On the deterministic interior body of random polytopes

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of independent copies of a random vector $X$ in $\mathbb{R}^n$. We revisit the question to determine the asymptotic shape of the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$ where $N>n$. We show that for any $β\in (0,1)$ there exists a constant $c(β)>0$ such that the following holds true: If $μ$ is a Borel probability measure on ${\mathbb R}^n$ then, for all $N\geq c(β)n$ we have that $K_N\supseteq T_{β\ln(\frac{N}{n})}(μ)$ with probability greater than $1-\exp(-\tfrac{1}{2}N^{1-β}n^β)$, where $T_p(μ)$ is the convex set of all points $x\in\mathbb{R}^n$ with half-space depth greater than or equal to $e^{-p}$. Our approach does not require any additional assumptions about the measure $μ$ and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family $\{T_p(μ)\}_{p>0}$ to other natural families of convex bodies associated with $μ$, such as the $L_p$-centroid bodies of $μ$ or the level sets of the Cramér transform of $μ$, and use this information in order to estimate the size of a random $K_N$.

math.MG↗