On the addition of convex sets in the hyperbolic plane

Analogue to the definition $K + L := bigcup_{xin K}(x + L)$ of theMinkowski addition in the euclidean geometry it is proposed to define the(noncommutative) addition $K vdash L := bigcup_{0, leqsl, rho,leqsl,a(varphi),0,leqsl,varphi,<, 2pi}T_{rho}^{(varphi)}(L)$ for compact,convex and smoothly bounded sets K and L in the hyperbolic plane $Omega$ (Kleins 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 ofdirection $varphi$. In general this addition does not preserveconvexity but nevertheless we may prove as main results: (1) $o in$ int$K, o in$ int L and K,L horocyclic convex imply the strictconvexity of $K vdash L$, and (2) in this case there exists a hyperbolic mixedvolume $V_h(K,L)$ of K and L which has a representation by a suitableintegral over the unit circle.