Maybe a simpler problem could be tackled first. What is a line segment best at being best at, compared to all other curves with the same two endpoints? (“Given an ill-posed problem that you don’t know how to well-pose, find a simpler ill-posed problem that you don’t know how to well-pose, and then well-pose it.”) Jim On Sunday, July 15, 2018, Dan Asimov <dasimov@earthlink.net> wrote:

Fair enough. Each function

F : C_n —> R

depending only on shape* that takes its global maxima (or minima) exactly on all spheres, and the other extrema exactly on all regular simplices . . . is a *candidate* for characterizing the distinction between spheres and simplices as opposite types of convex bodies.

I am guessing that maybe there is just one geometric objective function like F that is somehow *at the root* of the gradient between spheres and simplices in C_n. This may be wrong, but it feels right to me.

—Dan ————— * something I omitted mention of below

Fred Lunnon wrote: ----- << And why is one better than another? >>

The question seems ill-posed without some information about the purposes for which such a flow might be "better" ?! WFL

Dan Asimov wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.)

(Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function

F : C_n —> R

there exists a global maximum and a global minimum on C_n.)

For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.)

But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex?

Say n = 3. We could use the isoperimetric inequality, and look at the boundary-area to volume ratio, or better yet the dimensionless ratio

F(c) = A(c)^3 / V(c)^2

for c in C_n. (By a theorem in analysis, the boundary of a convex body is rectifiable, so A makes sense.)

Or we could take the ratio of the radii of the inscribed and circumscribed spheres:

F(c) = inradius(c) / circumradius(c),

c in C_n.

Question: Is one of these better than the other, or is something else even better for characterizing the gradient between the sphere and the simplex? And why is one better than another? -----

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun