Reverse Alexandrov-Fenchel inequalities for zonoids

The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume V (K1,K2,K3, . . . ,Kn) of convex bodies K1, . . . ,Kn in R by the product of the mixed volumes V (K1,K1,K3, . . . ,Kn) and V (K2,K2,K3, . . . ,Kn). As a consequence, for integers α1, . . . , αm ∈ N with α1 + · · · + αm = n the product Vn(K1) α1 n · · ·Vn(Km) αm n of suitable powers of the volumes Vn(Ki) of the convex bodies Ki, i = 1, . . . ,m, is a lower bound for the mixed volume V (K1[α1], . . . ,Km[αm]), where αi is the multiplicity with which Ki appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume V (K1[α1], . . . ,Km[αm]) in terms of the product of the intrinsic volumes Vαi(Ki), for i = 1, . . . ,m. The case where m = 2, α1 = 1, α2 = n − 1 has recently been settled by the present authors (2020). The case where m = 3, α1 = α2 = 1, α3 = n− 2 has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that K2 is a zonoid and K3 is the Euclidean unit ball. The case where α2 = · · · = αm = 1, K1 is the unit ball and K2, . . . ,Km are zonoids has been considered by Hug, Schneider (2011). 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.