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The classical Brunn-Minkowski theory studies the geometry of convex bodies inEuclidean space by use of the Minkowski sum. It originated from H. Brunn'sthesis in 1887 and H. Minkowski's paper in 1903. Because there is nouniversally acknowledged definition of the sum of two sets in hyperbolic space,there has been no Brunn-Minkowski theory in hyperbolic space since 1903. Inthis 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 theoryin the hyperbolic space by use of our hyperbolic $p$-sum, and we call it thehorospherical $p$-Brunn-Minkowski theory. Let $K$ be any smooth horosphericallyconvex bounded domain in the hyperbolic space $\mathbb{H}^{n+1}$. Throughcalculating the variation of the $k$-th modified quermassintegral of $K$ by useof our hyperbolic $p$-sum, we introduce the horospherical $k$-th $p$-surfacearea measure associated with $K$ on the unit sphere $\mathbb{S}^n$. For $k=0$,we introduce the horospherical $p$-Minkowski problem, which is the prescribedhorospherical $p$-surface area measure problem. Through designing and studyinga new volume preserving flow, we solve the existence of solutions to thehorospherical $p$-Minkowski problem for all $p \in (-\infty,+\infty)$ when thegiven measure is even. For $1 \leq k \leq n-1$, 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 thehorospherical $p$-Christoffel-Minkowski problem for $p\in(-n, +\infty)$ underappropriate assumption on the given measure. We also study the Brunn-Minkowskiinequalities and the Minkowski inequalities for domains in the hyperbolicspace.

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