Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $B$?

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.