Unless I have misunderstood something, the only functions satisfying your first constraint are constant --- consider a simplex and its circumsphere, which share n+1 points? WFL On 7/16/18, 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? -----
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