PreprintCC BY 4.0arXiv
The equality case in Godbersen's inequality and its generalizations
Abstract
Godbersen's long-standing conjecture states that the mixed volumes of any convex body K ⊂ ℝn satisfy V(K[k], −K[n−k])≤ binomnk vol(K) for any k ∈ {1, …, n − 1}, with equality if and only if K is an n-simplex. Kotrbatý and Mouamine recently proved the inequality in full generality, and showed that if K is a (full-dimensional) polytope attaining equality, it must be a simplex. We complete the characterization of the equality case: if K ⊂ ℝn is a convex body for which V(K[k], −K[n−k]) = binomnk vol(K) for some k ∈ {1, …, n −1}, then K is an n-simplex. By a similar method, we characterize the equality case in the higher-order Godbersen inequality conjectured by Schneider and by Kotrbatý.
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BibTeX
@misc{friedland2026equality,
title = {{The equality case in Godbersen's inequality and its generalizations}},
author = {Omer Friedland and Eli Putterman and Maud Szusterman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04527},
eprint = {2610.04527},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04527},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The equality case in Godbersen's inequality and its generalizations
AU - Omer Friedland
AU - Eli Putterman
AU - Maud Szusterman
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.04527
UR - https://arxiv.org/abs/2610.04527
AB - Godbersen's long-standing conjecture states that the mixed volumes of any convex body K ⊂ ℝ^n satisfy V(K[k], −K[n−k])≤ binomnk vol(K) for any k ∈ {1, …, n − 1}, with equality if and only if K is an n-simplex. Kotrbatý and Mouamine recently proved the inequality in full generality, and showed that if K is a (full-dimensional) polytope attaining equality, it must be a simplex. We complete the characterization of the equality case: if K ⊂ ℝ^n is a convex body for which V(K[k], −K[n−k]) = binomnk vol(K) for some k ∈ {1, …, n −1}, then K is an n-simplex. By a similar method, we characterize the equality case in the higher-order Godbersen inequality conjectured by Schneider and by Kotrbatý.
ER -