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In [SZ], Soprunov and Zvavitch have translated the Bezout inequalities (fromAlgebraic Geometry) into inequalities of mixed volumes satisfied by thesimplex. They conjecture this set of inequalities characterizes the simplex,among all convex bodies in R^n. Together with Saroglou, they proved thecharacterization among all polytopes [SSZ1] and, for a larger set ofinequalities, among all convex bodies [SSZ2]. The conjecture remains open for n\geq 4. In this work, we investigate necessary conditions on the structure ofthe boundary of a convex body K, for K to satisfy all inequalities. Inparticular, we obtain a new solution of the 3-dimensional case.

Extemizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes