PreprintCC BY-NC-ND 4.0HAL
On the equality case in the Kotrbatý-Mouamine inequality
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. Kotrbat\'y 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, resolving Godbersen's conjecture in the affirmative: 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.
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BibTeX
@misc{friedland2026equality,
title = {{On the equality case in the Kotrbatý-Mouamine inequality}},
author = {Omer Friedland and Eli Putterman and Maud Szusterman},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05776242v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the equality case in the Kotrbatý-Mouamine inequality
AU - Omer Friedland
AU - Eli Putterman
AU - Maud Szusterman
PY - 2026
DA - 2026/10/02
PB - HAL
UR - https://hal.science/hal-05776242v1
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. Kotrbat\'y 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, resolving Godbersen's conjecture in the affirmative: 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.
ER -