[math-fun] Relative convexity

Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"? Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! Jim Propp