On the addition of convex sets in the hyperbolic plane

Abstract

Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the Minkowski addition in the euclidean geometry it is proposed to define the (noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho’\,\leqsl\, a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho’}^{(\varphi)}(L)$ for compact, convex and smoothly bounded sets K and L in the hyperbolic plane $\Omega$ (Klein’s model). Here $\rho = a(\varphi)$ is the representation of the boundary $\partial$ K in geodesic polar coordinates and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length $\rho$ along the line through the origin o of direction $\varphi$. In general this addition does not preserve convexity but nevertheless we may prove as main results: (1) $o \in$ int $K, o \in$ int L and K,L “horocyclic convex” imply the strict convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed volume $V_h(K,L)$ of K and L which has a representation by a suitable integral over the unit circle.