PreprintarXiv licencearXiv
A note on consequences of Mahler's inequalities
Abstract
We derive Makai's inequality voln(K)voln((K−K)∘)≥(n+1)/n! in every dimension from the symmetric Mahler theorem in one higher dimension, with equality precisely for simplices. The proof uses a classical two-layer construction, an identity for its polar volume, and a Rogers--Shephard volume estimate. We then record basic consequences of the Mahler and Makai inequalities for polar successive minima, successive minima at the Santaló point, critical determinants, first covering minima, and non-separable lattice arrangements. We also combine Alexandrov--Fenchel with the volume-product inequalities to obtain sharp bounds for paired mixed volumes.
Related
- A Note on Fractional Sobolev inequalities revisited
- Addendum to: Mahler's method in several variables and finite automata
- Finding New Limit Points of Mahler Measure by Methods of Missing Data Restoration
- Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$
- A relation between Mahler volume and flag number for convex polytopes
BibTeX
@misc{lian2026note,
title = {{A note on consequences of Mahler's inequalities}},
author = {Yanlu Lian and Fei Xue},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34396},
eprint = {2610.34396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34396},
}
RIS
TY - UNPB
TI - A note on consequences of Mahler's inequalities
AU - Yanlu Lian
AU - Fei Xue
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34396
UR - https://arxiv.org/abs/2610.34396
AB - We derive Makai's inequality vol_n(K)vol_n((K−K)^∘)≥(n+1)/n! in every dimension from the symmetric Mahler theorem in one higher dimension, with equality precisely for simplices. The proof uses a classical two-layer construction, an identity for its polar volume, and a Rogers--Shephard volume estimate. We then record basic consequences of the Mahler and Makai inequalities for polar successive minima, successive minima at the Santaló point, critical determinants, first covering minima, and non-separable lattice arrangements. We also combine Alexandrov--Fenchel with the volume-product inequalities to obtain sharp bounds for paired mixed volumes.
ER -