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Bezout Inequality for Mixed volumes
Soprunov, Ivan...

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Bezout Inequality for Mixed volumes by Soprunov, Ivan ( Author )
Lemonjar Open Access
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02-07-2015
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.
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https://arxiv.org/abs/1507.00765
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