Horocyclic Brunn-Minkowski inequality

Given two non-empty subsets $A$ and $B$ of the hyperbolic plane ${\mathbb{H}}^2$, we define their horocyclic Minkowski sum with parameter $\lambda =1\mathrm{/}2$ as the set $[A:B{]}_{1\mathrm{/}2}\subseteq {\mathbb{H}}^2$ 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.