I am looking for a good measure of the spread of a set.
For example, a cylinder C is more spread out than a sphere of the same volume. So one guess is using diam(A). But in the above example , I could instead use two crossed-cylinders with less diameter than diam(C) and total volume equal to C. So I want a measure of the spread of a set in various distinct directions.
Say we have two sets A,B with equal volume and surface area, is it possible for ,say, A to be contained in a sphere that B can't up to rigid translation?
If not, then the ratio surface area/ volume is a good measure of the spread of a set.
Thoughts
The capacity of the set is a good candidate for measuring spread (http://en.wikipedia.org/wiki/Capacity_of_a_set) but I would still like a more easily computable/geometric definition.