Okay. What I've decided is the most relevant distinction between the simplex and the sphere in R^ is that the simplex is the convex hull of only n+1 points, whereas the sphere is the convex hull of a set of points that must be dense in the sphere. I'm sure I could have expressed the question better. And I still hope to someday. —Dan ----- 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? ... ... ----- -----