A contribution to equality in Alexandrov-Fenchel's inequality (Q1336192)

For convex bodies \(K,L,C_ 1, \dots, C_{n - 2}\) in Euclidean \(n\)- space, the Aleksandrov-Fenchel inequality states that \[ V(K,L, {\mathcal C})^ 2 \geq V(K,K, {\mathcal C}) V(L,L, {\mathcal C}), \tag{1} \] where \({\mathcal C} = (C_ 1, \dots, C_{n - 2})\). The conditions for equality are unknown in the general case. The authors consider the special case where \(K,L\) are polytopes, \(C_ 1, \dots, C_{n - 2}\) are polytopes lying in parallel hyperplanes with normal vector \(w\), and (without loss of generality) \(V(K,K, {\mathcal C}) = V(L,L, {\mathcal C}) \neq 0\). They prove that equality holds in (1) if and only if the bodies \(K + F(L,w)\) and \(L + F(K,w)\) have the same \((C_ 1, \dots, C_{n - 2})\)-extreme supporting hyperplanes. Here \(F(K,w)\) is the face of \(K\) with outer normal vector \(w\), and a hyperplane with normal vector \(u\) is called \((C_ 1, \dots, C_{n - 2})\)-extreme if each face \(F(C_ i,u)\) contains a segment \(S_ i\) so that \(\dim (S_ 1 + \cdots + S_{n - 2}) = n - 2\). The proof uses various deformations of polytopes and is not easy to follow. \{Reviewer's remark. A short proof of a more general result, where \(K,L\) may be arbitrary convex bodies, was recently given by the reviewer [In: ``Polytopes: abstract, convex and computational'' (eds. T. Bisztriczky et al.), NATO ASI Ser., Ser. C, Math. Phys. Sci. 440, 273-299 (1994)]\}.