Isoperimetric dimension for any (metric) measure space?

Question

$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.

$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$

for all open with smooth boundary $D\subset M$, differentiable manifold $M$, and universal constant $C$ for $M$. But it has been defined for graphs as well.

Q1: Let's start with metric measure space $(X,\Sigma, \mu, d)$, what are the minimal conditions and objects we need to add to it, to have the notion of isoperimetric dimension?

I think having just a measure space $(X,\Sigma, \mu)$ is not enough because we need a metric to define a perimeter length eg. via Minkoswki $\displaystyle \lim_{\varepsilon\to 0} \frac{\mu(A_{\varepsilon})-\mu(A)}{\varepsilon}$.

Q2: In other words, what is an example of a measure space $(X,\Sigma, \mu)$, and different metrics $d_1,d_2$ , each $(X,\Sigma, \mu, d_i)$ having different isoperimetric dimensions $\dim_1,\dim_2$

Finally,

Q3: Is there any counterexample for metric measure spaces, where we can't define isoperimetric dimension?

Probably some counterexample where there is no universal constant $C$ for any $d\in [0,\infty]$.

And preferably a continuum example. Because for example in the metric measure space $(\mathbb{Z}^{2},d_{discrete}, \mu_{counting })$ , sets are boundariless, but I think it is natural that the perimeter of say a 2D discrete square $Q$ is the number of edges between $Q$ and $Q^{c}$.

Whereas in the continuum I am more comfortable with perimeter of a set being

$$\displaystyle \lim_{\varepsilon\to 0} \frac{\mu(A_{\varepsilon})-\mu(A)}{\varepsilon}.$$

Answer 1

The main problem in defining isoperimetric inequalities on general measure metric spaces is that in the classical manifold case one actually has to deal not just with the original "volume" measure on state space, but also with a completely different "area" measure on appropriately defined "codimension 1 surfaces", which is not a part of the original definition of a measure metric space. One way to bypass this difficulty consists in trying to mimick the classical differential geometry in the absence of a manifold structure by extending, in a rather involved way, the notion of currents. One disadvantage though is that this approach does not work for the simplest measure metric spaces there are, namely, for countable graphs.

An alternative approach is based on the observation that in order to talk about isoperimetric inequalities it might be enough just to agree, for any subset of the state space, on what the size of its boundary is without necessarily specifying the boundary itself. This approach was implemented by Kaimanovich who introduced the isoperimetric inequalities for a general measure space endowed with a Markov operator (i.e., just a collection of transition probabilities associated to points of the state space). No explicit metric is required, although in natural situations the Markov operator can be defined in terms of a metric on the state space (e.g., the simple random walk in the case of graphs, or the Brownian motion in the case of Riemannian manifolds).

The classical theorems on the equivalence of isoperimetric inequalities and the embeddings of the associated Sobolev spaces (see the fundamental book of Maz'ya on Sobolev spaces) then carry over to this extended setup as well.

PS The Wikipedia entry on isoperimetric dimension contains a number of completely misleading mistakes. To begin with, there is no way to define it for a finite graph.

Answer 2

There is a notion of currents (as in Federer-Fleming currents from geometric measure theory) in general metric spaces, due to Ambrosio-Kirchheim (as well as a related one due to Urs Lang).

In some papers of Stefan Wenger (see here and others), he uses these to define and discuss isoperimetric inequalities in quite general metric spaces. Of course this is only one way of doing it, but maybe this can be helpful to you.

As far as your Q2 and Q3 there are probably many things to say, but as a first attempt how do you feel about "isoperimetric" inequalities on $\mathbb{R}$ with Lebesgue measure and the metric $d(x,y) = |x-y|^{1/2}$?