A contribution to equality in Alexandrov-Fenchel's inequality

Abstract

Since more than half a century it is an unsolved problem to characterize equality in Alexandrov-Fenchel's inequality

$$V(K, L, K_1 ,..., K_{n - 2} )^2 \geqslant V(K, K, K_1 ,..., K_{n - 2} )V(L, L, K_1 ,..., K_{n - 2} ),$$

, for convex bodiesK, L, K ₁, ...,K _n−2, examples being known in whichK, L are not homothetic. We present a solution in caseK, L, K ₁, ...,K _n−2 are polytopes andK ₁, ...,K _n−2 lie in a hyperplane which they span affinely.