Ashort proof of Alexandrov-Fenchel\'s inequality

Abstract

More than half a century ago Alexandrov [1] and Fenchel [8] proved a generalization of Minkowski's inequalities on volume and surface area of convex bodies: Let $K, L, K₁,\ldots,K_{n-2}$ be convex bodies in $Rⁿ$, and let $V( ·,\ldots,·)$ denote mixed volume. Then (AF) $$V(K,L,K₁,\ldots, K_{n-2})² ≥ V(K,K,K₁,\ldots, K_{n-2})V(L,L,K₁,\ldots, K_{n-2})$$ (For proofs see also Busemann [4], and Leichtweiss [9]). New interest in (AF)has been stimulated recently, partly by the discovery of its equivalence with the Hodge inequality in case of compact projective toric varieties (see Teissier [13], Khovanskij in Burago-Zalgaller [3]).The problem of characterizing equality in (AF) is still unsolved, though progress has been made during the last five years by R. Schneider ([10], [11], [12]),E. Tondorf, and the author ([5], [6], [7]. The method we have introduced hereby in [5] has meanwhile turned out to be applicable to a short and relatively elementary proof of (AF); we present it in this note. We are hopeful it will also contributed to a better understanding of (AF) and open problems connected with the inequality.