Horocyclic Brunn-Minkowski inequality

Given two non-empty subsets A and B of the hyperbolic plane H-2, we define their horocyclic Minkowski sum with parameter lambda = 1/2 as the set [A : B](1/2) C 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. The horocyclic Minkowski sum with parameter 0 < lambda < 1 is defined analogously. We prove that when A and B are Borel-measurable, root Area([A : B](lambda))>= (1 - lambda) Area(A) + lambda Area(B),where Area stands for hyperbolic area, with equality when A and B are concentric discs in the hyperbolic plane. We also give horocyclic versions of the Pr & eacute;kopa-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. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article studies the horocyclic Minkowski sum of two subsets in the hyperbolic plane and its properties. It proves an inequality relating the area of the subsets when they are Borel-measurable, and provides a connection to other inequalities.