On Godbersen's Conjecture

2014/08/09 by Shiri Artstein-Avidan, Keshet Einhorn, Artstein-Avidan, S. +5
Mathematics · #26b25 #52a20 #52a39 #52a40 #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1408.2135

openalex publication_date 2014/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a natural generalization of a geometric conjecture of Fáry and Rédei regarding the volume of the convex hull of K ⊂ \mathbb Rⁿ, and its negative image -K. We show that it implies Godbersen's conjecture regarding the mixed volumes of the convex bodies K and -K. We then use the same type of reasoning to produce the currently best known upper bound for the mixed volumes V(K[j], -K[n-j]), which is not far from Godbersen's conjectured bound. To this end we prove a certain functional inequality generalizing Colesanti's difference function inequality.

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