{"title":"Borell's inequality and mean width of random polytopes via discrete inequalities","authors":"David Alonso-Gutiérrez, Luis C. García-Lirola","doi":"arxiv-2407.18235","DOIUrl":null,"url":null,"abstract":"Borell's inequality states the existence of a positive absolute constant\n$C>0$ such that for every $1\\leq p\\leq q$ $$ \\left(\\mathbb E|\\langle X,\ne_n\\rangle|^p\\right)^\\frac{1}{p}\\leq\\left(\\mathbb E|\\langle X,\ne_n\\rangle|^q\\right)^\\frac{1}{q}\\leq C\\frac{q}{p}\\left(\\mathbb E|\\langle X,\ne_n\\rangle|^p\\right)^\\frac{1}{p}, $$ whenever $X$ is a random vector uniformly\ndistributed in any convex body $K\\subseteq\\mathbb R^n$ containing the origin in\nits interior and $(e_i)_{i=1}^n$ is the standard canonical basis in $\\mathbb\nR^n$. In this paper, we will prove a discrete version of this inequality, which\nwill hold whenever $X$ is a random vector uniformly distributed on\n$K\\cap\\mathbb Z^n$ for any convex body $K\\subseteq\\mathbb R^n$ containing the\norigin in its interior. We will also make use of such discrete version to\nobtain discrete inequalities from which we can recover the estimate $\\mathbb E\nw(K_N)\\sim w(Z_{\\log N}(K))$ for any convex body $K$ containing the origin in\nits interior, where $K_N$ is the centrally symmetric random polytope\n$K_N=\\operatorname{conv}\\{\\pm X_1,\\ldots,\\pm X_N\\}$ generated by independent\nrandom vectors uniformly distributed on $K$ and $w(\\cdot)$ denotes the mean\nwidth.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"47 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.18235","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Borell's inequality states the existence of a positive absolute constant
$C>0$ such that for every $1\leq p\leq q$ $$ \left(\mathbb E|\langle X,
e_n\rangle|^p\right)^\frac{1}{p}\leq\left(\mathbb E|\langle X,
e_n\rangle|^q\right)^\frac{1}{q}\leq C\frac{q}{p}\left(\mathbb E|\langle X,
e_n\rangle|^p\right)^\frac{1}{p}, $$ whenever $X$ is a random vector uniformly
distributed in any convex body $K\subseteq\mathbb R^n$ containing the origin in
its interior and $(e_i)_{i=1}^n$ is the standard canonical basis in $\mathbb
R^n$. In this paper, we will prove a discrete version of this inequality, which
will hold whenever $X$ is a random vector uniformly distributed on
$K\cap\mathbb Z^n$ for any convex body $K\subseteq\mathbb R^n$ containing the
origin in its interior. We will also make use of such discrete version to
obtain discrete inequalities from which we can recover the estimate $\mathbb E
w(K_N)\sim w(Z_{\log N}(K))$ for any convex body $K$ containing the origin in
its interior, where $K_N$ is the centrally symmetric random polytope
$K_N=\operatorname{conv}\{\pm X_1,\ldots,\pm X_N\}$ generated by independent
random vectors uniformly distributed on $K$ and $w(\cdot)$ denotes the mean
width.