arXiv subjects

Minas Pafis

Publications and source records attributed to Minas Pafis.

9 recordsLinked to original sources

arXiv2026

Cram\'er transform, half-space depth and threshold phenomena for convex bodies

We study the relationship between the Cram\'er transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure $\mu_K$ on a convex body $K\subseteq\mathbb{R}^n$, we prove the sharp pointwise comparison $$\Lambda_K^*(x)\leq -\log q_K(x)\leq \Lambda_K^*(x)+\frac12\log n+C,\qquad x\in\operatorname{int}(K),$$ where $C$ is an absolute constant. The order $\log n$ is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cram\'er transform. As consequences, we obtain sharp-order moment and tail estimates for $\Lambda_K^*$ and identify $\exp(\Lambda_K^*(x))$, up to polynomial factors in the dimension, with the number of independent samples needed for $x$ to be captured by their random convex hull. We also establish an $O(n^2)$ variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on $\ell_p$-balls for every $p>1$. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.

math.PR↗

arXiv2026

Many Facets in Random Polytopes from Product and Log-Concave Measures

We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $\mu$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $\mu([x^\ast-s,x^\ast])\asymp s^\kappa$ near its right endpoint $x^\ast$. For every sufficiently small fixed $\alpha>0$, the convex hull of $N=\lfloor e^{\alpha n}\rfloor$ independent points with law $\mu^{\otimes n}$ has at least $n^{n/2}e^{-C_{\mu,\alpha}n}$ expected facets; this includes all symmetric finite-alphabet distributions. For every full-dimensional log-concave probability measure on $\R^n$, we prove that there exist $T\in[n,2n]$ and $N=\lceil e^Tn^{3/2}\rceil$ for which \[ n^{n/2}e^{-Cn} \leq \mathbb E f_{n-1}(P_N) \leq n^{n/2}e^{Cn}. \] Thus the scale $n^{n/2}$, up to exponential factors, is universal for log-concave measures in this high-dimensional exponential regime. Finally, we construct a symmetric isotropic full-support non-log-concave counterexample with only $(1+o(1))2^n$ expected facets.

math.PR↗

arXiv2026

Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes

We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on $\mathbb{R}^d$ with full support. Our results extend the corresponding Gaussian lower bounds of B\'ar\'any and Vu and of B\'ar\'any and Th\"ale to this broader class of distributions. Our proof builds on the geometric construction introduced by B\'ar\'any and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.

math.PR↗

arXiv2026

Geometry of the subgaussian body of an isotropic convex body

For a centered convex body $K\subset\mathbb{R}^n$, let $\Psi_2(K)$ denote the symmetric convex body whose support function is given by the $\psi_2$-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 $\Psi_2(K)$ 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

Discrete log-concavity and threshold phenomena for atomic measures

We investigate threshold phenomena for random polytopes $K_N=\conv\{X_1,\dots,X_N\}$ generated by i.i.d.\ samples from an atomic law $\mu$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--F\"uredi--McDiarmid, where the supporting half-space estimate is derived via a smooth (gradient/uniqueness) step that can fail at boundary contact points. We then compare threshold-driving mechanisms in the continuous log-concave setting -- through the Cram\'{e}r transform and Tukey's half-space depth -- with their discrete analogues. Within this framework, we establish a sharp threshold for lattice $p$-balls $\mathbb{Z}^n \cap rB_p^n$. Finally, we present structural counterexamples showing that sharp thresholds need not hold in general discrete log-concave settings.

math.PR↗

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 $\beta\in (0,1)$ there exists a constant $c(\beta)>0$ such that the following holds true: If $\mu $ is a Borel probability measure on ${\mathbb R}^n$ then, for all $N\geq c(\beta)n$ we have that $K_N\supseteq T_{\beta\ln(\frac{N}{n})}(\mu)$ with probability greater than $1-\exp(-\tfrac{1}{2}N^{1-\beta}n^{\beta})$, where $T_p(\mu)$ 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 $\mu$ and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family $\{T_p(\mu)\}_{p>0}$ to other natural families of convex bodies associated with $\mu$, such as the $L_p$-centroid bodies of $\mu$ or the level sets of the Cram\'{e}r transform of $\mu$, and use this information in order to estimate the size of a random $K_N$.

math.MG↗

arXiv2023

Threshold for the expected measure of the convex hull of random points with independent coordinates

Let $\mu$ be an even Borel probability measure on ${\mathbb R}$. For every $N>n$ consider $N$ independent random vectors $\vec{X}_1,\ldots ,\vec{X}_N$ in ${\mathbb R}^n$, with independent coordinates having distribution $\mu $. We establish a sharp threshold for the product measure $\mu_n$ of the random polytope $K_N:={\rm conv}\bigl\{\vec{X}_1,\ldots,\vec{X}_N\bigr\}$ in ${\mathbb R}^n$ under the assumption that the Legendre transform $\Lambda_{\mu}^{\ast}$ of the logarithmic moment generating function of $\mu$ satisfies the condition $$\lim\limits_{x\uparrow x^{\ast}}\dfrac{-\ln \mu ([x,\infty ))}{\Lambda_{\mu}^{\ast}(x)}=1,$$ where $x^{\ast}=\sup\{x\in\mathbb{R}\colon \mu([x,\infty))>0\}$. An application is a sharp threshold for the case of the product measure $\nu_p^n=\nu_p^{\otimes n}$, $p\geq 1$ with density $(2\gamma_p)^{-n}\exp(-\|x\|_p^p)$, where $\|\cdot\|_p$ is the $\ell_p^n$-norm and $\gamma_p=\Gamma(1+1/p)$.

math.PR↗

arXiv2022

Threshold for the expected measure of random polytopes

Let $\mu$ be a log-concave probability measure on ${\mathbb R}^n$ and for any $N>n$ consider the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$, where $X_1,X_2,\ldots $ are independent random points in ${\mathbb R}^n$ distributed according to $\mu $. We study the question if there exists a threshold for the expected measure of $K_N$. Our approach is based on the Cramer transform $\Lambda_{\mu}^{\ast }$ of $\mu $. We examine the existence of moments of all orders for $\Lambda_{\mu}^{\ast }$ and establish, under some conditions, a sharp threshold for the expectation ${\mathbb E}_{\mu^N}[\mu (K_N)]$ of the measure of $K_N$: it is close to $0$ if $\ln N\ll {\mathbb E}_{\mu }(\Lambda_{\mu}^{\ast })$ and close to $1$ if $\ln N\gg {\mathbb E}_{\mu }(\Lambda_{\mu}^{\ast })$. The main condition is that the parameter $\beta(\mu)={\rm Var}_{\mu }(\Lambda_{\mu}^{\ast })/({\mathbb E}_{\mu }(\Lambda_{\mu }^{\ast }))^2$ should be small.

math.PR↗

arXiv2022

Half-space depth of log-concave probability measures

Given a probability measure $\mu $ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $\varphi_{\mu }(x)=\inf\{\mu (H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of ${\mathbb R}^n$ containing $x$. We show that if $\mu $ is log-concave then $$e^{-c_1n}\leq \int_{\mathbb{R}^n}\varphi_{\mu }(x)\,d\mu(x) \leq e^{-c_2n/L_{\mu}^2}$$ where $L_{\mu }$ is the isotropic constant of $\mu $ and $c_1,c_2>0$ are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of $L_q$-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

math.PR↗