Higher-order Godbersen inequalities and their equality cases

In his 1938 thesis, C. Godbersen conjectured that, among convex bodies of fixed positive volume, the mixed volume of a body and its reflection about the origin is maximized by simplices. Although not stated explicitly in Godbersen's thesis, it has been commonly conjectured that simplices are the only maximizers. Previously, only special cases and consequences of this conjecture were known. The former include convex bodies in dimensions two and three, bodies of constant width, and locally anti-blocking bodies. The latter include, for example, the Rogers-Shephard inequality for the volume of the difference body.

In this talk we will present a full proof of Godbersen's conjecture, including the characterization of the equality cases. The proof of the inequality is motivated by a more general conjecture proposed in 2000 by R. Schneider, in which the body and its reflection are replaced by an array of natural embeddings of the body into a product space. The proof of equality cases uses a new characterization of simplices in terms of their face lattice. Moreover, our method extends naturally to this higher-order setting, providing a full proof of Schneider's conjecture.

Based on joint work with F. Fryš, M. A. Mouamine, and S. Wang.