Geodesics and generalized geodesics

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Introduction

There are many ways that we can describe Wasserstein metric. One of them is to characterize absolutely continuos curves (AC)(p.188[1]) and provide a dynamic formulation of the special case Namely, it is possible to see as an infimum of the lengts of curves that satisfy Continuity equation.

Geodesics in general metric spaces

First, we will introduce definition of the geodesic in general metric space . In the following sections. we are going to follow a presentation from the book by Santambrogio[1] with some digression, here and there.

Definition. A curve is said to be geodesic in if it minimizes the length among all the curves
such that and .

Since we have a definition of a geodesic in the general space, it is natural to think of Riemannian structure. It can be defined. More about this topic can be seen in the following article Formal Riemannian Structure of the Wasserstein_metric.

Now, we proceed with necessary definitions in order to be able to understand Wasserstein metric in a different way.

Definition. A metric space is called a length space if it holds
                    

A space is called geodesic space if the distance is attained for some curve .

Definition. In a length space, a curve is said to be constant speed geodesic between and in if it satisfies
                     for all 

It is clear that constant-speed geodesic curve connecting and is a geodesic curve. This is very important definition since,for , we have that every constant-speed geodesic is also in where almost everywhere in .
In addition, minimum of the set is attained by our constant-speed geodesic curve

For more information on constant-speed geodesics, especially how they depend on uniqueness of the plan that is induced by transport and characterization of a constant-speed geodesic look at the book by L.Ambrosio, N.Gilgi, G.Savaré [2] or the book by Santambrogio[1].

Statement of Theorem

Finally, we can rephrase Wasserstein metrics in dynamic language as mentioned in the Introduction.

Whenever is convex set and , is a geodesic space. Proof can be found in the book by Santambrogio[1].

Theorem.[1] Let . Then
      

In special case, when is compact, infimum is attained by some constant-speed geodesic.

Generalized geodesics

There are many ways to generalize this fact. We will talk about a special case and a displacement convexity. Here we follow again book by Santambrogio[3].

In general, the functional is not a displacement convex. We can fix this by introducing a generalized geodesic.

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References