PreprintCC BY 4.0arXiv
A relation between Mahler volume and flag number for convex polytopes
Abstract
Given a convex polytope P⊂Bbb Rd with ℱ(P) many flags, we prove vol(P) vol(P−P)∘ ≤ ℱ(P)/(d!)2. This implies the following relation between Mahler volume and the number of flag conjectured by Freij, Schmitt, Schymura and Ziegler: for a centrally symmetric polytope P⊂Bbb Rd holds vol(P)vol(P∘) ≤ 2d/(d!)2 ℱ(P). This shows that the Mahler conjecture implies Kalai's flag conjecture.
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BibTeX
@misc{winter2026relation,
title = {{A relation between Mahler volume and flag number for convex polytopes}},
author = {Martin Winter},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09990},
eprint = {2610.09990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09990},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A relation between Mahler volume and flag number for convex polytopes
AU - Martin Winter
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09990
UR - https://arxiv.org/abs/2610.09990
AB - Given a convex polytope P⊂Bbb R^d with ℱ(P) many flags, we prove vol(P) vol(P−P)^∘ ≤ ℱ(P)/(d!)^2. This implies the following relation between Mahler volume and the number of flag conjectured by Freij, Schmitt, Schymura and Ziegler: for a centrally symmetric polytope P⊂Bbb R^d holds vol(P)vol(P^∘) ≤ 2^d/(d!)^2 ℱ(P). This shows that the Mahler conjecture implies Kalai's flag conjecture.
ER -