On the addition of convex sets in the hyperbolic plane (Q1430540)

An operation of fundamental role in the theory of convex sets is Minkowski's addition which may be defined by \(K+L=\{x+y \mid x\in K, y\in L\}=\bigcup_{x\in K}(x+L)\) for arbitrary subsets \(K\) and \(L\) of the Euclidean \(n\)-space. A definition of a Minkowski sum in non-Euclidean geometries seems to be unknown. It is the aim of this paper to make an attempt for such a definition in the simplest case of the hyperbolic space of dimension 2 with constant curvature \(-1\) and compact smooth convex bodies \(K\) and \(L\). The definition proposed reads as follows: \[ K\vdash L=\bigcup_{ 0\leq\rho'\leq a(\varphi), 0\leq \varphi < 2\pi } T^{(\varphi)}_{\rho'}(L), \] where \(\rho=a(\varphi)\) is the representation of the boundary \(\partial K\) in geodesic polar coordinates and \(T^{(\varphi)}_{\rho'}\) is the hyperbolic translation of length \(\rho'\) along the line through the origin \(o\) of direction \(\varphi\). The author proves that, in general, this hyperbolic Minkowski addition ``\(\vdash\)'' is preserving compactness but it fails to be commutative or to preserve convexity or polytopality. Moreover, he shows that this addition depends on the choice of a special point ``\(o\)'' as ``origin''. Nevertheless, the author shows that more stringent assumptions for convex bodies \(K\) and \(L\) imply the convexity of their sum \(K\vdash L\). The main results are restricted to the case \(o\in\text{int\,} K\), \(o\in\text{int\,} L\) and \(K\), \(L\) horocyclic convex and read as follows: (1) \(K\vdash L\) is strictly convex; (2) 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; (3) in the special case when \(L\) is the ball \(B_\varepsilon (o)\) about \(o\) of radius \(\varepsilon\geq 0\), Kohlmann's formula \[ \begin{aligned} V(K\vdash B_\varepsilon (o)) & = V(K)+U(K)\sinh\varepsilon +(V(K)+2\pi)(\cosh\varepsilon -1)\\ & =V(K)+\biggl(U(K)\frac{U(B_\varepsilon (o))}{2\pi} +V(K)\frac{V(B_\varepsilon (o))}{2\pi}\biggr) +V(B_\varepsilon (o)) \end{aligned} \] is proven, which is similar to Steiner's formula. Here \(V\) is area and \(U\) is perimeter.