Reverse Alexandrov–Fenchel inequalities for zonoids

Abstract

The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume [Formula: see text] of convex bodies [Formula: see text] in [Formula: see text] by the product of the mixed volumes [Formula: see text] and [Formula: see text]. As a consequence, for integers [Formula: see text] with [Formula: see text] the product [Formula: see text] of suitable powers of the volumes [Formula: see text] of the convex bodies [Formula: see text], [Formula: see text], is a lower bound for the mixed volume [Formula: see text], where [Formula: see text] is the multiplicity with which [Formula: see text] appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume [Formula: see text] in terms of the product of the intrinsic volumes [Formula: see text], for [Formula: see text]. The case where [Formula: see text], [Formula: see text], [Formula: see text] has recently been settled by the present authors (2020). The case where [Formula: see text], [Formula: see text], [Formula: see text] has been treated by Artstein-Avidan et al. under the assumption that [Formula: see text] is a zonoid and [Formula: see text] is the Euclidean unit ball. The case where [Formula: see text], [Formula: see text] is the unit ball and [Formula: see text] are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.