On the inequality for volume and Minkowskian thickness

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.