Stability of the reverse Blaschke–Santaló 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⊂Rn 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(Cn), where Cn 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(Cn). This result is then applied to strengthen a uniqueness result in stochastic geometry.