[Paper Review] Minimal volume product near Hanner polytopes

Jaegil Kim|arXiv (Cornell University)|Dec 11, 2012

Point processes and geometric inequalities25 references3 citations

TL;DR

This paper proves that every Hanner polytope is a strict local minimizer of the volume product among symmetric convex bodies in $ℝ^n$ under the Banach-Mazur distance. Using geometric perturbation analysis and volume comparisons with associated simplices, it establishes a linear lower bound on the volume product increase near Hanner polytopes, confirming their local optimality in Mahler's conjecture.

ABSTRACT

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed dimension. It is known that every Hanner polytope has the same volume product as the cube or the cross-polytope. In this paper we prove that every Hanner polytope is a strict local minimizer for the volume product in the class of symmetric convex bodies endowed with the Banach-Mazur distance.

Motivation & Objective

• To establish the local minimality of Hanner polytopes for the volume product in the class of symmetric convex bodies.
• To extend the local minimality result beyond the cube, which was previously known, to the broader class of Hanner polytopes.
• To provide a quantitative estimate of how much the volume product increases when a symmetric convex body deviates from a Hanner polytope.
• To use geometric and analytic techniques to compare volumes of perturbed bodies and their polars near Hanner polytopes.

Proposed method

• Analyzes the volume product $\mathcal{P}(K) = |K||K^\circ|$ for symmetric convex bodies $K$ near a Hanner polytope $H$ using the Banach-Mazur distance.
• Employs a perturbation argument based on the Hausdorff distance $\delta = \mathrm{d}_{\mathcal{H}}(K,H)$ to compare $K$ with a star-shaped polytope $P = \bigcup_{\mathbb{F}} X_{\mathbb{F}}$ derived from the flag structure of $H$.
• Uses volume comparison between $K$ and $P$, and between $K^\circ$ and $P^\circ$, to derive lower bounds on $|K||K^\circ|$ in terms of $\delta$.
• Applies the invariance of the volume product under linear transformations and polarity to fix a position and simplify the analysis.
• Relies on the structure of Hanner polytopes as $\ell_1$ or $\ell_\infty$ sums of lower-dimensional Hanner polytopes to define associated simplices $X_{\mathbb{F}}$ and their polars $X^*_{\mathbb{F}^*}$.
• Establishes a linear lower bound $\mathcal{P}(K) \geq \mathcal{P}(H) + c(n)\delta$ by combining volume increments from perturbations and known volume identities for Hanner polytopes.

Experimental results

Research questions

• RQ1 Is every Hanner polytope a strict local minimizer of the volume product in the class of symmetric convex bodies under the Banach-Mazur distance?
• RQ2 How does the volume product change when a symmetric convex body $K$ is perturbed away from a Hanner polytope $H$?
• RQ3 Can a quantitative lower bound on the increase of the volume product be established in terms of the Banach-Mazur or Hausdorff distance to $H$?
• RQ4 Does the geometric structure of Hanner polytopes allow for a stable volume comparison argument that confirms local minimality?

Key findings

• Every Hanner polytope is a strict local minimizer of the volume product in the class of symmetric convex bodies under the Banach-Mazur distance.
• For any symmetric convex body $K$ sufficiently close to a Hanner polytope $H$, the volume product satisfies $\mathcal{P}(K) \geq \mathcal{P}(H) + c(n)\delta$, where $\delta = \mathrm{d}_{\mathcal{BM}}(K,H) - 1$ and $c(n) > 0$ depends only on dimension $n$.
• The constant $c(n)$ is explicitly bounded below by $\frac{2^{n-1}\varepsilon^2}{n!^3}$, derived from the volume of the unit cross-polytope and combinatorial factors.
• The proof establishes that the volume product increases linearly with the distance $\delta$ from $H$, confirming strict local minimality.
• The result extends the local minimality of the cube (previously shown by Nazarov et al.) to the entire class of Hanner polytopes.
• The analysis relies on comparing $K$ and $K^\circ$ to a star-shaped polytope $P = \bigcup_{\mathbb{F}} X_{\mathbb{F}}$ associated with the flag structure of $H$, and uses volume increments from perturbations.