Stability of the reverse Blaschke-Santalo inequality for zonoids and applications

An important GL(n) invariant functional of centred (origin symmetric) convex bodies that has received particular attention is the volume product. For a centred convex body A subset of R-n it is defined by P(A) := vertical bar A vertical bar . vertical bar A*vertical bar. where vertical bar.vertical bar denotes volume and A* is the polar body of A. If A is a centred zonoid, then it is known that P(A) >= P(C-n), where C-n is a centred affine cube. i.e. a Minkowski sum of n linearly independent centred segments. Equality holds in the class of centred zonoids if and only if A is a centred affine cube. Here we sharpen this uniqueness statement in terms of a stability result by showing in a quantitative form that the Banach-Mazur distance of a centred zonoid A from a centred affine cube is small if P(A) is close to P(C-n). This result is then applied to strengthen a uniqueness result in stochastic geometry. (C) 2009 Elsevier Inc. All rights reserved.