On the Inequality for Volume and Minkowskian Thickness

Canadian mathematical bulletin

关键词: convex body;    geometric inequality;    thickness;    Minkowski space;    Banach space;    normed space;    reduced body;    Banach-Mazur compactum;    (modified) Banach-Mazur distance;    volume ratio;   

DOI  :  10.4153/CMB-2006-019-4

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Given a centrally symmetric convex body $B$ in $E^d,$ we denoteby $M^d(B)$ the Minkowski space ({em i.e.,} finite dimensionalBanach space) with unit ball $B.$ Let $K$ be an arbitrary convexbody in $M^d(B).$ The relationship between volume $V(K)$ and theMinkowskian thickness ($=$ minimal width) $hns_B(K)$ of $K$ cannaturally be given by the sharp geometric inequality $V(K) gealpha(B) cdot hns_B(K)^d,$ where $alpha(B)>0.$ As a simplecorollary of the Rogers--Shephard inequality we obtain that$inom{2d}{d}{}^{-1} le alpha(B)/V(B) le 2^{-d}$ with equalityon the left attained if and only if $B$ is the difference body ofa simplex and on the right if $B$ is a cross-polytope. The mainresult of this paper is that for $d=2$ the equality on the rightimplies that $B$ is a parallelogram. The obtained results yieldthe sharp upper bound for the modified Banach--Mazur distance to theregular hexagon.

【 授权许可】

Unknown