This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y. In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p". Jim Jim On Sat, Oct 24, 2020 at 12:22 PM Dan Asimov <dasimov@earthlink.net> wrote:

Yes. For any smooth metric surface (or n-dimensional manifold) M, a subset X of M is "geodesically convex" if for any points p, q of X there is a unique shortest geodesic curve C in M connecting p and q, and C lies entirely in X.

—Dan

Jim Propp wrote: ----- 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! -----

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