Hyperbolic $p$-sum and Horospherical $p$-Brunn-Minkowski theory in hyperbolic space

Abstract

The classical Brunn-Minkowski theory studies the geometry of convex bodies in Euclidean space by use of the Minkowski sum. It originated from H. Brunn's thesis in 1887 and H. Minkowski's paper in 1903. Because there is no universally acknowledged definition of the sum of two sets in hyperbolic space, there has been no Brunn-Minkowski theory in hyperbolic space since 1903. In this paper, for any $p>0$ we introduce a sum of two sets in hyperbolic space, and we call it the hyperbolic $p$-sum. Then we develop a Brunn-Minkowski theory in the hyperbolic space by use of our hyperbolic $p$-sum, and we call it the horospherical $p$-Brunn-Minkowski theory. Let $K$ be any smooth horospherically convex bounded domain in the hyperbolic space ${\mathbb{H}}^{n+1}$. Through calculating the variation of the $k$-th modified quermassintegral of $K$ by use of our hyperbolic $p$-sum, we introduce the horospherical $k$-th $p$-surface area measure associated with $K$ on the unit sphere ${\mathbb{S}}^n$. For $k=0$, we introduce the horospherical $p$-Minkowski problem, which is the prescribed horospherical $p$-surface area measure problem. Through designing and studying a new volume preserving flow, we solve the existence of solutions to the horospherical $p$-Minkowski problem for all $p\in (-\infty ,+\infty )$ when the given measure is even. For $1\le k\le n-1$, we introduce the horospherical $p$-Christoffel-Minkowski problem, which is the prescribed horospherical $k$-th $p$-surface area measure problem. We solve the existence of solutions to the horospherical $p$-Christoffel-Minkowski problem for $p\in (-n,+\infty )$ under appropriate assumption on the given measure. We also study the Brunn-Minkowski inequalities and the Minkowski inequalities for domains in the hyperbolic space.

156 pages, 7 figures. Comments are welcome