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Isotropic constants and regular polytopes

Christian Kipp

TL;DR

The paper develops first-order optimality conditions for the isotropic constant $L_K$ and integrates RS-movements to extract sharp structural information about polytopal extremals. It proves a strong rigidity phenomenon: a polytopal local maximizer with a simplicial vertex must be a simplex, and a centrally symmetric local maximizer with a simplicial vertex must be a cross-polytope, with analogous results for zonotopes containing cubical zones yielding cubes. In the zonotope setting, the authors show that extrema within the class of zonotopes with at most $n+1$ generators exist, with the maximum achieved only by affine images of the cube and the minimum by highly symmetric polytopes $Q_n$; these findings reinforce the deep link between first-order conditions, RS-movements, and affine symmetries in determining extremal shapes. Overall, the results illustrate how local geometric symmetries and affine reflectors constrain polytopal extremals of the isotropic constant, supporting the broader isotropic constant program through structural rigidity.

Abstract

We discuss first-order optimality conditions for the isotropic constant and combine them with RS-movements to obtain structural information about polytopal maximizers. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer with a simplicial vertex must be a simplex. A similar statement is shown for a centrally symmetric local maximizer with a simplicial vertex: it has to be a cross-polytope. Moreover, we show that a zonotope that maximizes the isotropic constant and that has a cubical zone must be a cube. Finally, we consider the class of zonotopes with at most n+1 generators and determine the extremals in this class.

Isotropic constants and regular polytopes

TL;DR

The paper develops first-order optimality conditions for the isotropic constant and integrates RS-movements to extract sharp structural information about polytopal extremals. It proves a strong rigidity phenomenon: a polytopal local maximizer with a simplicial vertex must be a simplex, and a centrally symmetric local maximizer with a simplicial vertex must be a cross-polytope, with analogous results for zonotopes containing cubical zones yielding cubes. In the zonotope setting, the authors show that extrema within the class of zonotopes with at most generators exist, with the maximum achieved only by affine images of the cube and the minimum by highly symmetric polytopes ; these findings reinforce the deep link between first-order conditions, RS-movements, and affine symmetries in determining extremal shapes. Overall, the results illustrate how local geometric symmetries and affine reflectors constrain polytopal extremals of the isotropic constant, supporting the broader isotropic constant program through structural rigidity.

Abstract

We discuss first-order optimality conditions for the isotropic constant and combine them with RS-movements to obtain structural information about polytopal maximizers. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer with a simplicial vertex must be a simplex. A similar statement is shown for a centrally symmetric local maximizer with a simplicial vertex: it has to be a cross-polytope. Moreover, we show that a zonotope that maximizes the isotropic constant and that has a cubical zone must be a cube. Finally, we consider the class of zonotopes with at most n+1 generators and determine the extremals in this class.
Paper Structure (6 sections, 20 theorems, 53 equations, 4 figures)

This paper contains 6 sections, 20 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. First-order conditions for extremals of the isotropic constant
  3. Simplicial vertices
  4. The centrally symmetric case
  5. Cubical zones of zonotopes
  6. Zonotopes with at most $n+1$ generators

Key Result

Theorem 1.1

Let $P \subset \mathbb{R}^n$ be a polytope that locally maximizes the isotropic constant. If $P$ is simplicial, then $P$ is a simplex.

Figures (4)

  • Figure 1: The settings of Theorem \ref{['thm_simplicial_vertex']}, Theorem \ref{['thm_symmetric_simplicial_vertex']} and Theorem \ref{['thm_cubical_zone']}, respectively. We show that local maximizers of $K \mapsto L_K$ with these properties have to be a simplex, a cross-polytope and a cube, respectively.
  • Figure 2: The RS-movement $(P_t)_{i \in [-\varepsilon,\varepsilon]}$. The vertex $v$ moves orthogonally to the linear hyperplane spanned by the ridge $G$ and the origin (marked in blue).
  • Figure 3: The SRS-movement $(P_t)_{i \in [-\varepsilon,\varepsilon]}$. The vertices $v$ and $-v$ move orthogonally to the linear hyperplane spanned by the ridge $G$ and the origin (marked in blue).
  • Figure 4: The SRS-movement $(Z_t)_{i \in I}$. The points $x$ and $y$ move along the red dashed line segments.

Theorems & Definitions (40)

  • Theorem 1.1: Rademacher
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 30 more