The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918500210049024 |
|---|---|
| author | Chen, Shibing Li, Yuanyuan Xi, Dongmeng Xu, Zhefeng |
| author_facet | Chen, Shibing Li, Yuanyuan Xi, Dongmeng Xu, Zhefeng |
| contents | The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body \(K=-K\subset \mathbb{R}^3\), \[
\VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] It was recently proved by Iriyeh--Shibata \cite{IS2020}, and a shorter proof was later given by Fradelizi--Hubard--Meyer--Roldán-Pensado--Zvavitch \cite{FHMRZ}. Both proofs combine ingenious equipartition arguments of algebraic-topological origin with delicate geometric estimates inspired by Meyer's argument for unconditional bodies.
In this paper, we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems. This is a natural extension of the new techniques developed in our proof of the three-dimensional non-symmetric Mahler conjecture \cite{CLXX-Mahler}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13795 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems Chen, Shibing Li, Yuanyuan Xi, Dongmeng Xu, Zhefeng Metric Geometry 52a20 The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body \(K=-K\subset \mathbb{R}^3\), \[ \VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] It was recently proved by Iriyeh--Shibata \cite{IS2020}, and a shorter proof was later given by Fradelizi--Hubard--Meyer--Roldán-Pensado--Zvavitch \cite{FHMRZ}. Both proofs combine ingenious equipartition arguments of algebraic-topological origin with delicate geometric estimates inspired by Meyer's argument for unconditional bodies. In this paper, we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems. This is a natural extension of the new techniques developed in our proof of the three-dimensional non-symmetric Mahler conjecture \cite{CLXX-Mahler}. |
| title | The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems |
| topic | Metric Geometry 52a20 |
| url | https://arxiv.org/abs/2605.13795 |