Curvdimension and curvature of a metric profile, part I

In the notes Sub-riemannian geometry from intrinsic viewpoint    I propose two notions related to the curvature of a metric space at one of its points: the curvdimension and the curvature of a metric profile.In this post I would like to explain in detail what is this about, as well as making a number of comments and suggestions which are not in the actual version of the notes.

Related to these notions, they stem from rather vague proposals first made in earlier papers Curvature of sub-riemannian spaces and Sub-riemannian geometry and Lie groups II.

I shall start with the definition of the metric profile associated to a point $x \in X$ of a locally compact metric space $(X,d)$.  We need first a short preparation.

Let $CMS$ be the collection of isometry classes of  pointed compact metric spaces.An element of $CMS$ is denoted like $[X,d,x]$ and is the equivalence class of a compact metric space $(X,d)$, with a specified point $x\in X$, with respect to the equivalence relation: two pointed compact metric spaces $(X,d,x)$, $(Y,D,y)$ are equivalent if there is a surjective  isometry $f: (X,d) \rightarrow (Y,D)$ such that $f(x) = y$.

The space $CMS$ is a metric space when endowed with the Gromov-Hausdorff distance between (isometry classes of) pointed compact metric spaces.

Definition 1.  Let $(X,d)$ be a locally compact metric space. The metric profile of $(X,d)$ at $x$ is the function which associates to $\varepsilon > 0$ the element of $CMS$ defined by $P^{m}(\varepsilon, x) = \left[\bar{B}(x,1), \frac{1}{\varepsilon} d, x\right]$

(defined for small enough $\varepsilon$, so that the closed metric ball $\bar{B}(x,\varepsilon)$ with respect to the distance $d$,  is compact).

Remark 1. See the previous post Example: Gromov-Hausdorff distance and the Heisenberg group, part II , where the behaviour of the metric profile of the physicists Heisenberg group is discussed.

The metric profile of the space at a point is therefore  a curve in another metric space, namely $CMS$ with a Gromov-Hausdorff distance. It is not any curve, but one which has certain properties which can be expresses with the help of the GH distance. Very intriguing, what about a dynamic induced along these curves in the $CMS$. Nothing is known about this, strangely!

Indeed, to any element $[X,d,x]$ of $CMS$ it is associated the curve $P^{m}(\varepsilon,x)$. This curve could be renamed $P^{m}(\varepsilon , [X,d,x])$.  Notice that $P^{m}(1 , [X,d,x]) = [X,d,x]$.

For a fixed $\varepsilon \in (0,1]$, take now $P^{m}(\varepsilon , [X,d,x])$, what is the metric profile of this element of $CMS$? The answer is: for any $\mu \in (0,1]$ we have $P^{m}(\mu , P^{m}(\varepsilon , [X,d,x])) = P^{m}(\varepsilon \mu , [X,d,x])$

which proves that the curves in $CMS$ which are metric profiles are not just any curves.

Definition 2. If the metric profile $P^{m}(\varepsilon ,[X,d,x])$ can be extended by continuity to $\varepsilon = 0$, then the space $(X,d)$ admits a metric tangent space at $x \in X$ and the isometry class of (the unit ball in) the tangent space equals $P^{m}(0 , [X,d,x])$.

You see, $P^{m}(0 , [X,d,x])$ cannot be any point from $CMS$. It has to be the isometry class of a metric cone, namely a point of $CMS$ which has constant metric profile.

The curvdimension and curvature explain how the the metric profile curve behaves near $\varepsilon = 0$. This is for the next post.

Curvature and Brunn-Minkowski inequality

A beautiful paper by Yann Ollivier and Cedric Villani

A curved BRUNN–MINKOWSKI INEQUALITY on the discrete hypercube OR: WHAT IS THE RICCI CURVATURE OF THE DISCRETE  HYPERCUBE?

The Brunn-Minkowski inequality  says that  the log  of the volume (in euclidean spaces) is concave. The concavity inequality is improved, in riemannian manifolds with Ricci curvature at least K, by a quadratic term with coefficient proportional with K.

The paper is remarkable in many ways. In particular are compared two roads towards curvature in spaces more general than riemannian: the coarse curvature introduced by Ollivier and the other based on the displacement convexity of the entropy function (Felix Otto , Cedric Villani, John Lott, Karl-Theodor Sturm), studied by many researchers. Both are related to  Wasserstein distances . NONE works for sub-riemannian spaces, which is very very interesting.

In few words, here is the description of the coarse Ricci curvature: take an epsilon and consider the application from the metric space (riemannian manifold, say) to the space of probabilities which associates to a point from the metric space the restriction of the volume measure on the epsilon-ball centered in that point (normalized to give a probability). If this application is Lipschitz with constant L(epsilon) (on the space of probabilities take the L^1 Wassertein distance) then the epsilon-coarse Ricci curvature times epsilon square is equal to 1 minus L(epsilon) (thus we get a lower bound of the Ricci curvature function, if we are in a Riemannian manifold). Same definition works in a discrete space (this time epsilon is fixed).
The second definition of Ricci curvature comes from inverse engineering of the displacement convexity inequality discovered in many particular spaces. The downside of this definition is that is hard to “compute” it.

Initially, this second definition was related to the L^2 Wasserstein distance which,  according to Otto calculus, gives to the space of probabilities (in the L^2 frame) a structure of an infinite dimensional riemannian manifold.

Concerning the sub-riemannian spaces, in the first definition the said application cannot be Lipschitz and in the second definition there is (I think) a manifestation of the fact that we cannot put, in a metrically acceptable way, a sub-riemannian space into a riemannian-like one, even infinite dimensional.