Are there convexity generalizations that admit genus information?

For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can be joined by a line or a circle seems to work.

Is there a good definition that works in higher dimensions for which appropriate generalization of traditional convex geometry inequalities such as Brunn-Minkowski inequality can be given (A suitable notion of convex hull needs to be first defined).