The wording of Dan’s question reminds me of a problem I’ve sometimes mused about: for what sort of n-athlon (whose n constituent events would not necessarily be sports-related in the narrow sense) would I be the world champion? How big would n have to be? (Has anyone written about the meta-game of designing games one would excel at?) Jim Propp On Wednesday, July 18, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
I've had a few days to think about this. At this point, I'm pretty sure *what* I think is the number one distinction between the extremes of the regular simplex and the sphere, though I haven't yet convinced myself I know exactly *why* I believe this.
Meanwhile, if anyone else would like to express their opinion, I'd be curious to learn what others think about this.
—Dan
I 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? ... ... -----
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