2024/11/27 by Shiri Artstein-Avidan, Artstein-Avidan, Shiri, Eli Putterman +1
Mathematics · #52A39 #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2412.05308
openalex publication_date 2024/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
The longstanding Godbersen's conjecture states that for any convex body K ⊂ \mathbb Rn of volume 1 and any j ∈ \0, …, n\, the mixed volume Vj = V(K[j], -K[n - j]) is bounded by \binomnj, with equality if and only if K is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the Vj, arising from different geometric constructions, are bounded above by their values when one substitutes \binomnj for Vj, with equality if and only if K is a simplex. One of our results implies that for any K of volume 1 we have (1)/(n + 1) ∑j = 0n \binomnj-1 Vj ≤ 1, showing that Godbersen's conjecture holds ''on average'' for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body.