Horocyclic Brunn-Minkowski inequality

TL;DR

This paper establishes a new Brunn-Minkowski type inequality for horocyclic Minkowski sums in hyperbolic space, along with related Prékopa-Leindler and Borell-Brascamp-Lieb inequalities, extending classical convex geometry results to hyperbolic geometry.

Contribution

It introduces a novel horocyclic Minkowski sum and proves associated inequalities, extending Brunn-Minkowski theory to hyperbolic geometry with new curvature considerations.

Findings

Proved a Brunn-Minkowski inequality for horocyclic Minkowski sums in hyperbolic space.

Derived horocyclic versions of Prékopa-Leindler and Borell-Brascamp-Lieb inequalities.

Equality holds for concentric discs in the hyperbolic plane.

Abstract

Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $\lambda=1/2$ 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 $0 < \lambda <1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_{\lambda} )} \geq (1-\lambda) \cdot \sqrt{ Area(A) } + \lambda \cdot \sqrt{ Area(B) }, $$ 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 Pr\'ekopa-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.