arXiv subjects

Antonios Hmadi

Publications and source records attributed to Antonios Hmadi.

5 recordsLinked to original sources

arXiv2026

Online balancing of vectors with small coordinates

Let $v_1,\ldots,v_T\in B_2^m$ be fixed in advance and revealed sequentially, and assume that $\|v_t\|_\infty\leqslant d^{-1/2}$ for some $d\geqslant 1$ and every $1\leqslant t\leqslant T$. There are absolute constants $L,C,c>0$ and a randomized online signing such that $$\mathbb{P}\left\{\max_{k\leqslant T}\left\|\sum_{t=1}^k\varepsilon_t v_t\right\|_\infty>6L\right\} \leqslant CT\exp\left(-\frac{cd}{\ln^2(ed)}\right).$$ Consequently, constant prefix discrepancy holds with probability at least $1-\varepsilon$ once $d$ is at least $C\ln\frac{3T}{\varepsilon}\left[\ln\left(e+\ln\frac{3T}{\varepsilon}\right)\right]^2$. In particular, every fixed sequence of vectors $a_t\in[-1,1]^m$ with at most $d$ nonzero coordinates admits an online signing with prefix discrepancy $O(\sqrt d)$ and failure probability at most $CT\exp[-cd/\ln^2(ed)]$. We also prove a nonuniform version in which the failure probability depends on the individual parameters $d_t=\|v_t\|_\infty^{-2}$, and a lower bound showing that a universal constant prefix discrepancy is impossible when $d=o(\ln T)$. We identify the corresponding $\ln^2 d$ barrier for the compact-potential method and extend the argument to general symmetric target bodies admitting a quadratic smoothness estimate.

math.CO↗

arXiv2026

Moment comparisons, Sudakov inequalities and entropy of centroid bodies

Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $ϕ$ and every $q\geqslant 1$, $$ \|ϕ(G)\|_q\leqslant C\sqrt{\ln(en)+q}\,\|ϕ(X)\|_q,\qquad \|ϕ(X)\|_q\leqslant C\left(\sqrt{\ln(en)}+ψ(X)\sqrt q\right)\|ϕ(G)\|_q. $$ Applied to support functions, this gives the sharp worst case order $C\sqrt{\ln(en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. Combining, at each level of Latala's dyadic chain, the strongest of the three quantitative $L_p$-Sudakov estimates used here yields $$ \left(\mathbb{E}\|Y\|^p\right)^{1/p}\leqslant C\left(n^{1/4}\sqrt{\ln(en)}\,\ln(e+\ln(en))\,\mathbb{E}\|X\|+σ_p(Y)\right) $$ whenever the weak moments of $Y$ are dominated by those of $X$. In a second direction, we study the generalized dual Sudakov problem for the self generated metrics associated with $Z_r(X)^\circ$ and prove dimension free packing estimates for $Z_p(X)$. We also obtain mean norm estimates for centroid bodies, factorization through arbitrary symmetric convex bodies and an affine dimensional refinement.

math.MG↗

arXiv2026

Dimension-free cotype for isotropic log-concave random polytope spaces

Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}^n$ with common isotropic log-concave distribution $μ$ and set $P_{N,n}^μ:=\operatorname{conv}\{\pm X_i:1\leqslant i\leqslant N\}$. Assume that $N/n=γ\geqslant γ_0$ where $γ_0>1$ is an absolute constant. We prove that with probability at least $1-Cγ\exp(-c n^{1/4})$ every $k$-dimensional subspace $E$ of $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^μ})$ satisfies $d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant cγ^{-C}k^α$ for every $1\leqslant k\leqslant n$ where $c,C,α>0$ are absolute constants. Consequently, with the same probability, $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^μ})$ has cotype $q(γ)<\infty$ with cotype constant depending only on $γ$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $μ$. The proof adapts the deterministic coefficient scheme of Huang-Tikhomirov replacing the Gaussian estimates in their argument by estimates for isotropic log-concave random matrices. As an application, using the log-concave extension of Gluskin's theorem, we obtain a separable Banach space of finite cotype for which the Banach-Mazur diameter of its $k$-dimensional subspaces is of order $k$ and whose finite-dimensional building blocks are generated by isotropic log-concave random polytopes.

math.FA↗

arXiv2026

Banach-Mazur distances and basis constants of isotropic log-concave random spaces

We study the Banach-Mazur distance between random normed spaces generated by centrally symmetric random polytopes associated with isotropic log-concave measures in $\mathbb{R}^n$. We show that, in a wide range of parameters, if $x_1,\dots,x_m$ and $y_1,\dots,y_m$ are independent samples from an isotropic log-concave probability measure on $\mathbb{R}^n$, then the corresponding normed spaces $X_{B_m}$ and $Y_{A_m}$ generated by their absolute convex hulls satisfy, with high probability, $$d_{\rm BM}(X_{B_m},Y_{A_m}) \geqslant \frac{cn}{\ln(1+m/n)},$$ which is sharp in both $n$ and $m$ and recovers the extremal order $n$ when $m \approx n$. Our results extend Gluskin's theorem from the Gaussian setting to general isotropic log-concave measures, providing evidence for a universality phenomenon in the extremal geometry of the Banach-Mazur compactum. In addition, we investigate operator-theoretic properties of the associated random spaces and, as consequences, we derive sharp estimates for their basis constant and show that these random spaces are far from the class of spaces with a $1$-unconditional basis. The proofs combine probabilistic and geometric methods with recent advances related to Bourgain's slicing problem.

math.FA↗

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↗