Horocyclic Brunn-Minkowski inequality

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, $$\sqrt{Area([A:B{]}_{\lambda })}\ge (1-\lambda )\cdot \sqrt{Area(A)}+\lambda \cdot \sqrt{Area(B)},$$ 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 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.

36 pages, 2 figures. v5: minor changes. Final version, to appear in Advances in Mathematics. v4: The paper has undergone considerable reorganization. The order of the sections has changed, and the focus is now mostly on horocycles without reference to more general path spaces

References in corpus (1)

• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

Cited by in corpus (1)

• Brunn-Minkowski inequalities for sprays on surfaces