Reverse Alexandrov-Fenchel inequalities for zonoids

The Alexandrov–Fenchel inequality bounds from below the square of mixed volume V(K1,K2,K3,…,Kn) convex bodies K1,…,Kn in ℝn by product volumes V(K1,K1,K3,…,Kn) and V(K2,K2,K3,…,Kn). As a consequence, for integers α1,…,αm∈ℕ with α1+⋯+αm=n Vn(K1)α1n⋯Vn(Km)αmn suitable powers Vn(Ki) Ki, i=1,…,m, is lower bound V(K1[α1],…,Km[αm]), where αi multiplicity which Ki appears volume. It has been conjectured Betke Weil that there reverse inequality, is, sharp upper V(K1[α1],…,Km[αm]) terms intrinsic Vαi(Ki), i=1,…,m. case m=2, α1=1, α2=n−1 recently settled present authors (2020). m=3, α1=α2=1, α3=n−2 treated Artstein-Avidan et al. under assumption K2 zonoid K3 Euclidean unit ball. α2=⋯=αm=1, K1 ball K2,…,Km are zonoids considered Hug Schneider. Here, we substantially generalize these previous contributions, cases most zonoids, thus provide further evidence supporting inequality. equality all inequalities characterized. More generally, stronger stability results established as well.