  Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii
we study the following Bezout type inequality for mixed volumes $$
V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K). $$ We show
that the above inequality characterizes simplices, i.e. if $K$ is a convex body
satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset
{\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of
the proof is to study perturbations given by Wulff shapes. In particular, we
prove a new theorem on differentiability of the support function of the Wulff
shape, which is of independent interest.
  In addition, we study the Bezout inequality for mixed volumes introduced in
arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies
which is strictly larger than the set of all polytopes that are non-simplices.
We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly
indecomposable convex bodies.
