On the inequality for volume and Minkowskian thickness

被引:5
作者
Averkov, Gennadiy [1 ]
机构
[1] Tech Univ Chemnitz, Fak Math, D-09107 Chemnitz, Germany
来源
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES | 2006年 / 49卷 / 02期
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;
D O I
10.4153/CMB-2006-019-4
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Given a centrally symmetric convex body. Bin E-d, we denote by M-d(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ball B. Let K be an arbitrary convex body in M-d(B). The relationship between volume V(K) and the Minkowskian thickness (= minimal width) Delta(B)(K) of K can naturally be given by the sharp geometric inequality V(K) >= alpha(B) center dot Delta(B)(K)(d), where alpha(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain that ((2d)(d))(-1) <= alpha(B)/V(B) <= 2(-d) with equality on the left attained if and only if B is the difference body of a simplex and on the right if B is a cross-polytope. The main result of this paper is that for d = 2 the equality on the right implies that B is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach-Mazur distance to the regular hexagon.
引用
收藏
页码:185 / 195
页数:11
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