arXiv subjects

Nikos Skarmogiannis

Publications and source records attributed to Nikos Skarmogiannis.

4 recordsLinked to original sources

arXiv2022

On a multi-integral norm defined by weighted sums of log-concave random vectors

Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb R}^n$. We show that if $C$ is isotropic then \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s \leq c_1L_C(\log n)^5\,\sqrt{n}M(K)\|{\bf t}\|_2\end{equation*} for every $s\geq 1$ and ${\bf t}=(t_1,\ldots ,t_s)\in {\mathbb R}^s$, where $L_C$ is the isotropic constant of $C$ and $M(K):=\int_{S^{n-1}}\|ξ\|_Kdσ(ξ)$. This reduces a question of V.~Milman to the problem of estimating from above the parameter $M(K)$ of an isotropic convex body. The proof is based on an observation that combines results of Eldan, Lehec and Klartag on the slicing problem: If $μ$ is an isotropic log-concave probability measure on ${\mathbb R}^n$ then, for any centrally symmetric convex body $K$ in ${\mathbb R}^n$ we have that $$I_1(μ,K):=\int_{{\mathbb R}^n}\|x\|_K\,dμ(x)\leq c_2\sqrt{n}(\log n)^5\,M(K).$$ We illustrate the use of this inequality with further applications.

math.FA↗

arXiv2019

Affine quermassintegrals of random polytopes

A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ asks whether for every convex body $K$ in ${\mathbb R}^n$ and all $1\leqslant k\leqslant n$ $$Φ_{[k]}(K):={\rm vol}_n(K)^{-\frac{1}{n}}\left (\int_{G_{n,k}}{\rm vol}_k(P_F(K))^{-n}\,dν_{n,k}(F)\right )^{-\frac{1}{kn}}\leqslant c\sqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $Φ_{[k]}(K)$ when $K=B_1^n$, the unit ball of $\ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.

math.MG↗

arXiv2019

A note on norms of signed sums of vectors

Our starting point is an improved version of a result of D. Hajela related to a question of Komlós: we show that if $f(n)$ is a function such that $\lim\limits_{n\to\infty }f(n)=\infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every $n\geqslant n_0$ and any $S\subseteq \{-1,1\}^n$ with cardinality $|S|\leqslant 2^{n/f(n)}$ one can find orthonormal vectors $x_1,\ldots ,x_n\in {\mathbb R}^n$ that satisfy $$\|ε_1x_1+\cdots +ε_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)}$$ for all $(ε_1,\ldots ,ε_n)\in S$. We obtain analogous results in the case where $x_1,\ldots ,x_n$ are independent random points uniformly distributed in the Euclidean unit ball $B_2^n$ or any symmetric convex body, and the $\ell_{\infty }^n$-norm is replaced by an arbitrary norm on ${\mathbb R}^n$.

math.MG↗

arXiv2019

Norms of weighted sums of log-concave random vectors

Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s\end{equation*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to "randomized" vector balancing problems.

math.MG↗