Random polytopes and the volume-product of symmetric convex bodies.

PARALLEL SECTIONS HOMOTHETY BODIES WITH MINIMAL MAHLER VOLUME IN ℝn

“…; the equality case was discussed in [17,23]. When K is a zonoid (limits of finite Minkowski sums of line segments), Meyer and Reisner (see, e.g., [9,21,22]) proved that the same inequality holds, with equality if and only if K is an n-cube. For the case of polytopes with at most 2n + 2 vertices (or facets) (see, e.g., [2] for references), Lopez and Reisner [13] proved the inequality (1.1) for n ≤ 8 and the minimal bodies are characterized.…”

“…Recently, Lin and Leng [12] gave a new and intuitive proof of the inequality (1.1) in R 2 . Reisner (see, e.g., [9,21,22]) established the same inequality for a class of bodies that have a high degree of symmetry, known as zonoids. Inequality (1.1) was established by Saint Raymond [24] for bodies which are symmetric with respect to the coordinate hyperplanes.…”

ORIGIN-SYMMETRIC CONVEX BODIES WITH MINIMAL MAHLER VOLUME IN ℝ2

“…; the equality case, obtained for 1 − ∞ spaces, is discussed in [10] and [13]. When K is a zonoid (limits of finite Minkowski sums of line segments), it was proved by S. Reisner in [11], [12] and [5] that the same inequality holds, with equality if and only if K is an n-cube.…”

Volume Inequalities for Sets Associated With Convex Bodies

“…For n > 3 there are bodies, different than parallelotopes and their polars, for which (3) is an equality, as shown by Saint Raymond [30]. Reisner [27], [28] and Saint Raymond [30] proved (3) for special classes of convex bodies, namely for zonoids and for the affine images of convex sets symmetric with respect to the coordinate hyperplanes. A simpler proof of Reisner's result was given in [13].…”