Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Horocyclic Brunn-Minkowski inequality

Rotem Assouline, Bo'az Klartag

Abstract

Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $λ=1/2$ as the set $[A:B]_{1/2} \subseteq \mathbb{H}^2$ of all midpoints of horocycle curves connecting a point in $A$ with a point in $B$. These horocycle curves are parameterized by hyperbolic arclength, and the horocyclic Minkowski sum with parameter $0 < λ<1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_λ )} \geq (1-λ) \cdot \sqrt{ Area(A) } + λ\cdot \sqrt{ Area(B) }, $$ where $Area$ stands for hyperbolic area, with equality when $A$ and $B$ are concentric discs in the hyperbolic plane. We also prove horocyclic versions of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength.

Horocyclic Brunn-Minkowski inequality

Abstract

Given two non-empty subsets and of the hyperbolic plane , we define their horocyclic Minkowski sum with parameter as the set of all midpoints of horocycle curves connecting a point in with a point in . These horocycle curves are parameterized by hyperbolic arclength, and the horocyclic Minkowski sum with parameter is defined analogously. We prove that when and are Borel-measurable, where stands for hyperbolic area, with equality when and are concentric discs in the hyperbolic plane. We also prove horocyclic versions of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength.
Paper Structure (12 sections, 22 theorems, 149 equations, 2 figures)

This paper contains 12 sections, 22 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Finsler structures
  4. Horocycles in the hyperbolic plane
  5. Proof of the inequality via horocyclic needle decomposition
  6. Proof of Theorem \ref{['thm318']}
  7. Brunn-Minkowski for horocyclic needles
  8. Horocyclic needle decomposition
  9. Partition via a guiding function
  10. A needle decomposition theorem for Finsler surfaces
  11. Proof of Theorem \ref{['horodecomp']}
  12. Appendix: proof of Lemma \ref{['tanglemma']}

Key Result

Theorem 1.1

Let $A, B \subseteq \mathbb H^2$ be non-empty, Borel measurable sets. Then for any $0 < \lambda < 1$, where $Area$ stands for hyperbolic area in $\mathbb H^2$. When $A$ and $B$ are concentric discs, or when one of the sets is a singleton, equality holds in (eq_319).

Figures (2)

  • Figure 1: Horocyclic vs. geodesic Minkowski summation in the upper half plane model. The horocyclic $\frac{1}{2}$-Minkowski sum of the two discs, which is the sum of midpoints of oriented horocycles joining the disc on the left to the one on the right, is the blue set in the middle. The geodesic $\frac{1}{2}$-Minkowski sum, i.e. the set of all midpoints of geodesics joining two discs, is the orange at the top.
  • Figure 2: Horocycles joining two points $x,y \in \mathbb H^2$, with the arrows indicating the orientation.

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • Lemma 3.2
  • ...and 34 more