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Godbersen's conjecture and the Lp-Rogers-Shephard inequality

  • Jan Kotrbatý
  • Mohamed A. Mouamine
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Abstract

We prove that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the classical Rogers-Shephard inequality. We also prove that simplices are the only extremizers among convex polytopes. Finally, we use this inequality to prove an Lp-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any p∈(1,∞], the only extremizers are simplices with a vertex at the origin.

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Cite

BibTeXRIS
BibTeX
@misc{kotrbaty2026godbersens,
  title = {{Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality}},
  author = {Jan Kotrbatý and Mohamed A. Mouamine},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.20387},
  eprint = {2607.20387},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.20387},
}
RIS
TY  - UNPB
TI  - Godbersen's conjecture and the L_p-Rogers-Shephard inequality
AU  - Jan Kotrbatý
AU  - Mohamed A. Mouamine
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2607.20387
UR  - https://arxiv.org/abs/2607.20387
AB  - We prove that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the classical Rogers-Shephard inequality. We also prove that simplices are the only extremizers among convex polytopes. Finally, we use this inequality to prove an L_p-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any p∈(1,∞], the only extremizers are simplices with a vertex at the origin.
ER  -