I think this has something to do with convexity. The shape of this is like a star and is convex. Is this a necessary condition for a ball?Prove that the set is not a unit ball in any metric induced by a norm on .
My point is that the way you defined convexity can only be determined (visually since that is how you are describing it) in . If I asked you a similar question for the Banach space where is some arbitrary metric (or even topological) space I doubt you could graph it and tell me whether or not it's "convex"