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Silouanos Brazitikos

Publications and source records attributed to Silouanos Brazitikos.

At least 19 recordsLinked to original sources
arXiv2026

List decoding of toric codes

Polynomial evaluation codes hold a prominent place in coding theory. In this work, we study the problem of list decoding for a general class of polynomial evaluation codes, also known as Toric codes, that are defined for any given convex polytope P. Special cases, such as Reed-Solomon and Reed-Muller codes, have been studied extensively. We present a generalization of the Guruswami-Sudan algorithm that takes into account the geometry and the combinatorics of P and compute bounds for the decoding radius.

cs.IT↗
arXiv2026

Hermite-Fisher bounds and stability for min-entropy power inequalities

We derive explicit lower bounds for relative Fisher information by combining a variational principle with suitably orthogonalized Hermite-polynomial test functions. The resulting cumulant bounds are asymptotically sharp and yield lower bounds for Gaussian entropy deficits. We also establish quantitative versions of sharp min-entropy power inequalities in all dimensions. En route, we develop a stability result for Brzezinski's sharp bound for block sections of products of Euclidean balls, which may be of independent interest.

cs.IT↗
arXiv2026

Thin-Shell implies small-ball deviation via Gaussian tilts

We show that uniform thin-shell estimates for isotropic log-concave measures $μ$ on $\mathbb{R}^n$ yield precise and explicit deviation estimates for the Euclidean norm $|X|$ below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: $$ μ\left(|X|\le \sqrt{n}-s\right) \le \exp\left\{ -\frac{s^2}{4} \left(1+O\left(\frac{s}{\sqrt{n}}\right)\right) \right\} \qquad \forall\, s\in(0,\sqrt{n}). $$ In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when $μ$ has general covariance structure) of these deviation estimates is also obtained.

math.FA↗
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 $μ$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $μ([x^\ast-s,x^\ast])\asymp s^κ$ near its right endpoint $x^\ast$. For every sufficiently small fixed $α>0$, the convex hull of $N=\lfloor e^{αn}\rfloor$ independent points with law $μ^{\otimes n}$ has at least $n^{n/2}e^{-C_{μ,α}n}$ 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

A Deterministic Proof of the Slicing Theorem

We give a deterministic proof of Bourgain's slicing theorem. The proof uses neither stochastic localization nor stochastic calculus, does not rely on Guan's covariance bound, and avoids $M$-ellipsoids, Pisier's regularization machinery, and complex interpolation. Instead, the argument is based on a deterministic one-parameter quadratic perturbation of the measure, a covariance-survival principle, and a geometric analysis of $L_p$-centroid bodies.

math.FA↗
arXiv2026

Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position

Let $K \subset \mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \[ \int_K x \otimes x \, dμ_K(x) = \mathrm{Id}_n. \] We give deterministic geometric proofs of \[ M(K) \leq C \frac{\log(n)}{\sqrt{n}} \qquad \text{and} \qquad M^*(K) \leq C \sqrt{n} \, \log(n), \] where \[ M(K) = \int_{\mathbb{S}^{n-1}} \|θ\|_K \, dσ(θ), \qquad M^*(K) = \int_{\mathbb{S}^{n-1}} h_K(θ) \, dσ(θ). \] Combining both estimates yields \[ M(K) M^*(K) \leq C \log^2(n). \] The first proof uses a quadratic aggregate of dyadic centroid bodies. The second uses the analogous weighted aggregate of the Laplace bodies $p\{Λ_K \leq p\}^{\circ}$, which are equivalent to the centroid bodies by the work of Klartag and E. Milman. In both cases, curvature at each dyadic scale outside a subspace of codimension $O(p)$ leads, via the min--max principle, Legendre duality, and the spherical Laplacian, to the required estimate. The only high-dimensional input is the dimension-free small-ball consequence of the slicing theorem.

math.MG↗
arXiv2026

Restricted Hyperplane Sections of the Cross-Polytope and the Simplex

We give a new proof of Webb's theorem on maximal central hyperplane sections of the regular \(n\)-simplex \(Δ_n\), viewed in its standard embedding in \(\mathbb R^{n+1}\). A similar method also yields sharp maximal estimates for non-central sections of \(Δ^n\) whose distance \(d\) from the barycenter is small, namely $d< \sqrt{\frac{1}{(n+1)(2n+1)}}.$ Moreover, we obtain sharp volume estimates for central hyperplane sections of the cross-polytope \(B_1^n\) that pass through the barycenter of a facet.

math.MG↗
arXiv2026

On Sections of Convex Bodies in John's Position and of Generalised $B_p^n$ Balls

We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity. This lets us complement some earlier results of the first two named authors, as well as generalise the classical estimates of Meyer-Pajor and Koldobsky regarding extremal sections of $B_p^n$ balls to a broader family of norms induced by a John's decomposition of the identity in $\mathbb{R}^n$.

math.MG↗
arXiv2026

Moments of sums of exponentials, beyond CHS

We establish a sharp lower bound on the $L_p$-norm of sums of independent exponential random variables with fixed variance, for $p \geq 2$, thus extending Hunter's positivity theorem (1976) for completely homogeneous polynomials. We determine the exact regime of $p$ where such sums enjoy Schur-monotonicity.

math.PR↗
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↗
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 $μ$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--Füredi--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 Cramér 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

Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables

We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial $h_{2k}(a_1,\dots,a_n)$ is attained precisely at the half-plus/half-minus vector and we compute the optimal value in closed form. The proof combines algebraic properties of $h_{2k}$ with the probabilistic representation $k!\,h_k(a)=\mathbb{E}(\sum_{i=1}^n a_iX_i)^k$, where $X_1,\dots,X_n$ are i.i.d. standard exponential random variables with density $e^{-x}1_{x>0}$ and a combinatorial identity. This viewpoint further yields sharp upper and lower bounds for $\mathbb{E}|\sum_{i=1}^n a_iX_i|^{q}$ under natural constraints on the coefficients, including the spherical constraint $\sum a_i^2=1$ combined with the non-negative regime $a_i\ge0$, or the centred regime $\sum a_i=0$. Moreover, we determine the exact minimum of $h_{2k}$ on the $\ell_\infty$-sphere $S_\infty = \{a \in \mathbb{R}^n : \|a\|_\infty = 1\}$, which yields sharp norm comparison inequalities between the matrix norms induced by complete homogeneous symmetric polynomials and the classical operator and Schatten norms.

math.PR↗
arXiv2025

Sharp estimates for the Cramér transform of log-concave measures and geometric applications

We establish a new comparison between the Legendre transform of the cumulant generating function and the half-space depth of an arbitrary log-concave probability distribution on the real line, that carries on to the multidimensional setting. Combined with sharp estimates for the Cramér transform of rotationally invariant measures, we are led to some new phase-transition type results for the asymptotics of the expected measure of random polytopes. As a byproduct of our analysis, we address a question on the sharp exponential separability constant for log-concave distributions, in the symmetric case.

math.PR↗
arXiv2023

Half-space depth of log-concave probability measures

Given a probability measure $μ$ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $φ_{μ}(x)=\inf\{μ(H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of ${\mathbb R}^n$ containing $x$. We show that if $μ$ is log-concave then $$e^{-c_1n}\leq \int_{\mathbb{R}^n}φ_{μ}(x)\,dμ(x) \leq e^{-c_2n/L_μ^2}$$ where $L_{μ}$ is the isotropic constant of $μ$ 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↗
arXiv2023

Threshold for the expected measure of random polytopes

Let $μ$ 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 $μ$. We study the question if there exists a threshold for the expected measure of $K_N$. Our approach is based on the Cramer transform $Λ_μ^{\ast }$ of $μ$. We examine the existence of moments of all orders for $Λ_μ^{\ast }$ and establish, under some conditions, a sharp threshold for the expectation ${\mathbb E}_{μ^N}[μ(K_N)]$ of the measure of $K_N$: it is close to $0$ if $\ln N\ll {\mathbb E}_{μ}(Λ_μ^{\ast })$ and close to $1$ if $\ln N\gg {\mathbb E}_{μ}(Λ_μ^{\ast })$. The main condition is that the parameter $β(μ)={\rm Var}_{μ}(Λ_μ^{\ast })/({\mathbb E}_{μ}(Λ_{μ}^{\ast }))^2$ should be small.

math.PR↗