International Mathematics Research Notices
In this paper we consider the following analog of Bezout inequality for mixed volumes: V(P1,…,Pr,Δn−r)Vn(Δ)r−1≤∏i=1rV(Pi,Δn−1) for 2≤r≤n. We show that the above inequality is true when Δ is an n-dimensional simplex and P1,…,Pr are convex bodies in Rn. We conjecture that if the above inequality is true for all convex bodies P1,…,Pr, then Δ must be an n-dimensional simplex. We prove that if the above inequality is true for all convex bodies P1,…,Pr, then Δ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to Δ), which confirms the conjecture when Δ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.
Soprunov, Ivan and Zvavitch, Artem, "Bezout Inequality for Mixed Volumes" (2016). Mathematics Faculty Publications. 166.
This is a pre-copyedited, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Soprunov, Ivan and Artem Zvavitch. "Bezout Inequality for Mixed Volumes." International Mathematics Research Notices, vol. 2016, no. 23, 2016, pp. 7230-7252, doi:10.1093/imrn/rnv390. is available online at: https://academic.oup.com/imrn/article/2016/23/7230/2633471