Linear combinations of convex hypersurfaces in non-Euclidean geometry
The author defines the Minkowski sum of two arbitrary strongly convex bodies in the framework provided by the Riemannian geometry (\(n\)-dimensional spherical geometry). The approach of the same problem in the Weierstrass geometry (\(n\)-dimensional hyperbolic geometry) leads to satisfactory results in the restricted case of distance convex bodies having \(\bar{0}=(1,0,\dots,0)\) in its interior. The resulted convex bodies of this operation are described in terms of a support function, which is implicitly expressed in both cases. The sum operation is extended to convex combinations and, as a consequence, the arithmetic mean of two convex sets is discussed in hyperbolic geometry.
distance convex bodies in hyperbolic space
quasi support functions in spherical space
scientific article; zbMATH DE number 6010036
scientific article