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PreprintCC BY 4.0arXiv

Around higher-order Godbersen conjectures

  • Filip Fryš
  • Jan Kotrbatý
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Abstract

We consider two higher-order generalizations of the Godbersen conjecture for mixed volumes of convex bodies, which were first proposed by Schneider in 2000. First, we establish the conjectures in several special cases, in particular in low dimensions. Second, we prove that certain consequences of the conjectures hold. More precisely, we define a higher-order version of the unbalanced difference body and prove related weighted inequalities, generalizing previous results of Artstein-Avidan and Putterman. Finally, we introduce higher-order analogs of unbalanced joins of convex bodies considered by Artstein-Avidan, Einhorn, Florentin, and Ostrover, prove the corresponding volume bounds, and propose conjectures that interpolate between the higher-order Godbersen conjectures and a conjecture due to Fáry and Rédei.

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Cite

BibTeXRIS
BibTeX
@misc{frys2026around,
  title = {{Around higher-order Godbersen conjectures}},
  author = {Filip Fryš and Jan Kotrbatý},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08612},
  eprint = {2609.08612},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08612},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Around higher-order Godbersen conjectures
AU  - Filip Fryš
AU  - Jan Kotrbatý
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.08612
UR  - https://arxiv.org/abs/2609.08612
AB  - We consider two higher-order generalizations of the Godbersen conjecture for mixed volumes of convex bodies, which were first proposed by Schneider in 2000. First, we establish the conjectures in several special cases, in particular in low dimensions. Second, we prove that certain consequences of the conjectures hold. More precisely, we define a higher-order version of the unbalanced difference body and prove related weighted inequalities, generalizing previous results of Artstein-Avidan and Putterman. Finally, we introduce higher-order analogs of unbalanced joins of convex bodies considered by Artstein-Avidan, Einhorn, Florentin, and Ostrover, prove the corresponding volume bounds, and propose conjectures that interpolate between the higher-order Godbersen conjectures and a conjecture due to Fáry and Rédei.
ER  -