On the Inequality for Volume and Minkowskian Thickness

Abstract

AbstractGiven a centrally symmetric convex bodyBin, we denote by ℳd(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ballB. LetKbe an arbitrary convex body in ℳd(B). The relationship between volumeV(K) and the Minkowskian thickness (= minimal width) ΔB(K) ofKcan naturally be given by the sharp geometric inequalityV(K) ≥ α(B) · ΔB(K)d, where α(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain thatwith equality on the left attained if and only ifBis the difference body of a simplex and on the right ifBis a cross-polytope. The main result of this paper is that ford= 2 the equality on the right implies thatBis a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.