Minkowski addition of H-convex sets and related Helly-type theorems

A natural generalization of the usual convexity notion is the notion of H-convexity. In the first part of the paper we will completely answer the question when the Minkowski sum of H-convex sets is itself an H-convex set. Based on this we are able to present, in the second part, results in the spirit of combinatorial geometry, namely some Helly-type theorems for H-convex sets.