Stability of the reverse Blaschke–Santaló inequality for zonoids and applications

Abstract

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 ⊂ R n it is defined by P ( A ) : = | A | ⋅ | A ∗ | , where | ⋅ | 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.