For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $\mu$ on $\mathbb R^n$, all nonempty symmetric convex sets $K,L\subseteq\mathbb R^n$, and all $\lambda\in[0,1]$, $$ \mu(\lambda K+(1-\lambda)L)^{p_n} \ge \lambda\mu(K)^{p_n}+(1-\lambda)\mu(L)^{p_n}, $$ where $$ p_n\ge \frac{c}{n^2\ln n} $$ for some absolute constant $c>0$.
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 f...
For $n\ge1$, let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain $n$ pairwise distinct integers $a_1, a_2, \dots, a_n$ with $k \mid a_k$ for $1\le k\le n$, and define $h_{\mathbb P}(n)$ analogously for the primes at most $n$. We prove \[ F(n)\le n^{4/3}\exp\!\left(O\!\left(\frac{\log n}{\log\log...
Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $\alpha_\gamma(n)$ be the largest number for which \[ \gamma_n(\lambda K+(1-\lambda)L)^{\alpha_\gamma(n)} \ge \lambda\gamma_n(K)^{\alpha_\gamma(n)} +(1-\lambda)\gamma_n(L)^{\alpha_\gamma(n)} \] holds for all convex bodies $K,L\subset\ma...
For an $n$-dimensional convex body $K$, let $\theta_L(K)$ denote its lattice covering density, and let $\Theta_L^{\mathrm{conv}}(n)$ and $\Theta_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantitie...
Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ imply $\int_Kf\leq C_n\...
For every $2<p<\infty$ and every integer $r\ge2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le2^{Cr^2}$, $\gamma_2(T)\le Cr$, and $\gamma_2(\conv T)\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]...
Helen W. J. Zhang, Cheng-Dong Zhao· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
