Characterization of Simplices via the Bezout Inequality for Mixed volumes · arXivDesk1512.05442Dec 17, 20158 pages
Characterization of Simplices via the Bezout Inequality for Mixed volumes
Christos Saroglou, Ivan Soprunov, Artem Zvavitch
Abstract
We consider the following Bezout inequality for mixed volumes: V(K1,…,Kr,Δ[n−r])Vn(Δ)r−1≤i=1∏rV(Ki,Δ[n−1]) for 2≤r≤n.
It was shown previously that the inequality is true for any
-dimensional simplex
and any convex bodies
K1,…,Kr in
. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies
K1,…,Kr in
. In this paper we prove that this is indeed the case if we assume that
is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex
-polytopes. In addition, we show that if a body
satisfies the Bezout inequality for all bodies
K1,…,Kr then the boundary of
cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.