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? —Dan