Horocyclic Brunn-Minkowski inequality

Abstract

Given two non-empty subsets A and B of the hyperbolic plane H2, we define their horocyclic Minkowski sum with parameter λ=1/2 as the set [A:B]1/2⊆H2 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<λ<1 is defined analogously. We prove that when A and B are Borel-measurable,Area([A:B]λ)≥(1−λ)⋅Area(A)+λ⋅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 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.