# Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes.

From what I have read, it seems that most infinite dimensional objects have infinite dimensional measure of either $0$ or $1$, so comparing the ratio of their infinite dimensional "volumes" seems to be a futile exercise.

Any ideas about how to quantify how much bigger $A$ is than $B$?

-
Is there an algebraic structure on them? For example the Birkoff Polytope is the collection of all doubly stochastic matrices. If so perhaps you could use some type of function space norm to measure things. – Wintermute Jun 7 '13 at 19:51
To be honest, I'm not sure. $A$ and $B$ are defined as the convex hulls of two sets of points in $\mathbb{R}^{\infty}$, if that is helpful. – okj Jun 7 '13 at 19:54
What are the sets of points? – Wintermute Jun 7 '13 at 19:54
I have a generalized fourier series, and the points are different sets of coefficients (i.e. each point is the fourier representation of a function). So $B$ is a subset of the functions contained in $A$. – okj Jun 7 '13 at 20:12
Maybe check and see if the functions are $L^1$ or $L^2$. – Wintermute Jun 8 '13 at 21:52