Here’s a definition that turns out not to be quite what I wanted but seems somewhat natural. Assume C is compact. Imagine you can teleport from any point in C to any other; then the ordinary metric d on R^3 gives rise to a “scrunched” quotient metric d/C on R^3 (to get from x to y you walk from x to a point nearest to x in C, teleport to a point nearest to y in C, and then walk to y), and a notion of a geodesic in the quotient metric. We could say that a set S that contains C is convex relative to C if there is a d/C geodesic that stays within S. This definition has the feature that if C is a circle and S is an r-neighborhood of C, then S is convex relative to C for small values of r but not for r close to the radius of C. I can’t decide how I feel about this. Jim Propp On Sat, Oct 24, 2020 at 2:41 PM James Propp <jamespropp@gmail.com> wrote:
It's more of a pre-theoretic feeling. I have a sense I'll recognize the definition I'm intuiting when I see it.
(I appreciate y'all being kinder to me than the MathOverflow crowd would be for my asking such a vague question!)
Jim
On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- 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 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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