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$β$-Uniform Convexity and Divisible Domains

Amelia Pompilio

Abstract

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the exact constant $β$ is related to the regularity of the boundary.

$β$-Uniform Convexity and Divisible Domains

Abstract

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is -uniformly convex, where the exact constant is related to the regularity of the boundary.

Paper Structure

This paper contains 8 sections, 4 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Divisible Convex Sets
  4. $\alpha$-Hölder regularity and $\beta$-convexity
  5. p-Uniform Convexity
  6. Coordinate Set-Up
  7. Proof of the main theorem
  8. $\beta$-Uniform Convexity of the Hilbert metric?

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{RP}^n$ be a strictly convex divisible domain. Then for any $x_0 \in \Omega,$ the space $(\Omega, M(x_0, \cdot))$ is $\beta$-uniformly convex for some $\beta \in [2, \infty)$.

Figures (2)

  • Figure 1: Case 2
  • Figure 2: Case 3

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: see e.g.Benoist
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Benoist Benoist
  • Theorem 2.5: Benoist Benoist
  • Definition 2.6: Guichard Guichard
  • Definition 2.8: Suzuki Suzuki
  • Claim 3.1
  • ...and 5 more