We try to go a little further from Which convex solids have geodesics on the surface that lie entirely in a plane?
Definitions: The surface of a finite 3D convex body may be called a convex surface. Consider two points P and Q on a convex surface S. Define the uniformity ratio at P with reference to Q as $$\frac{\text{length of the shortest path lying on the surface S between P and Q with no other constraint}} {\text{length of the shortest path lying on S between P and Q that also lies entirely in a plane}}\ $$
- Obviously, the uniformity ratio has max value 1 for any point pair; Does this ratio have a lower bound? And if so, for which pair of points on which convex solid?
Further, define the local uniformity of S at P as the average value of the uniformity ratio at P with reference to Q as P is kept fixed and Q runs all over S.
It appears that if S is smooth and P is an umbilical point, the local uniformity of S at P has the highest possible value of 1. Our attempt is thus to quantify departure of points from umbilical (intuitively, uniform) nature.
Which convex surface S (not necessarily smooth) minimizes the average value of the local uniformity measured at all its points and what is this average value?
If we do not require S to be smooth, are there convex surfaces such that the highest value of local uniformity measured over all points on S is less than 1? If so, which S minimizes this max value of local uniformity?
For a given S, the point(s) with minimum uniformity may be called the most nonuniform point on S. If S can be any (non-smooth) convex surface, is there a lower bound on the local uniformity at the most nonuniform point on S?
