Introduction

We are interested in the following question: given a compact subset K of ℝd with positive reach, and a Hausdorff approximation P of this set, is it possible to approximate Federer's curvature measures of K (see [9] or Section 12.2.2 for a definition) from P only? A positive answer to this question has been given in [8] using convex analysis. In this chapter, we show that such a result can also be deduced from a careful study of the “size” – that is, the covering numbers – of the medial axis.

The notion of medial axis, also known as ambiguous locus in Riemannian geometry, has many applications in computer science. In image analysis and shape recognition, the skeleton of a shape is often used as an idealized version of the shape, which is known to have the same homotopy type as the original shape [14].

Overview

Since the creation of Ricci flow by Hamilton in 1982, a rich theory has been developed in order to understand the behaviour of the flow, and to analyse the singularities that may occur, and these developments have had profound applications, most famously to the Poincaré conjecture. At the heart of the theory lie a large number of a priori estimates and geometric constructions, which include most notably the Harnack estimates of Hamilton, the L-length of Perelman (in the spirit of Li-Yau), the logarithmic Sobolev inequality arising from Perelman's W-entropy, and the reduced volume of Perelman, amongst others.

The objective of these lectures is to explain this theory from the point of view of optimal transportation. As I explain in Section 5.4, Ricci flow and optimal transportation combine rather well, and we will see fundamental but elementary aspects of this when we see in Theorem 5.2 how diffusions contract under reverse-time Ricci flow. However, the key to the whole theory is to realise to which object one should apply this result: not the original Ricci flow, but a new Ricci flow derived from the original one, on a base manifold of one higher dimension, that we call the canonical soliton. In this way, essentially the entire foundational theory of Ricci flow mentioned above drops out naturally.

Throughout the lectures I emphasise the intuition; the objective is to demonstrate how one can discover the theory rather than treat it as a black box that just happens to work.

Introduction and main concepts

Lyapunov conditions appeared a long time ago. They were particularly well fitted to deal with the problem of convergence to equilibrium for Markov processes; see [23, 38–40] and references therein. They also appeared earlier in the study of large and moderate deviations for empirical functionals of Markov processes (for examples, see Donsker and Varadhan [21, 22], Kontoyaniis and Meyn [33, 34], Wu [47, 48], Guillin [28, 29]), for solving the Poisson equation [24].

Their use to obtain functional inequalities is however quite recent, even if one may afterwards find hint of such an approach in Deuschel and Stroock [19] or Kusuocka and Stroock [35]. The present authors and coauthors have developed a methodology that has been successful for various inequalities: Lyapunov–Poincaré inequalities [4], Poincaré inequalities [3], transportation inequalities for Kullback information [17] or Fisher information [32], super Poincaré inequalities [16], weighted and weak Poincaré inequalities [13], or [18] for super weighted Poincaré inequalities. We finally refer to the forthcoming book [15] for a complete review. For more references on the various inequalities introduced here we refer to [1, 2, 36, 46]. The goal of this short review is to explain the methodology used in these papers and to present various general sets of conditions for this panel of functional inequalities. The proofs will of course be only schemed and we will refer to the original papers for complete statements.

Introduction

These notes are based on a series of lectures given by the second author for the Summer School “Optimal Transportation: Theory and Applications” in Grenoble during the week of June 22–26, 2009.

We try to summarize some of the main results concerning gradient flows of geodesically λ-convex functionals in metric spaces and applications to diffusion partial differential equations (PDEs) in the Wasserstein space of probability measures. Due to obvious space constraints, the theory and the references presented here are largely incomplete and should be intended as an oversimplified presentation of a quickly evolving subject. We refer to the books [3, 68] for a detailed account of the large literature available on these topics.

In the Section 6.2 we collect some elementary and well-known results concerning gradient flows of smooth convex functions in ℝd. We selected just a few topics, which are well suited for a “metric” formulation and provide a useful guide for the more abstract developments. In the Section 6.3 we present the main (and strongest) notion of gradient flow in metric spaces characterized by the solution of a metric evolution variational inequality: the aim here is to show the consequence of this definition, without any assumptions on the space and on the functional (except completeness and lower semicontinuity); we shall see that solutions to evolution variational inequalities enjoy nice stability, asymptotic, and regularization properties.

Introduction

This chapter is extended notes based on the author's lecture series at the summer school at Université Joseph Fourier, Grenoble: “Optimal Transportation: Theory and Applications.” The aim of these five lectures (corresponding to Sections 7.3–7.7) was to review the recent impressive development on the interplay between optimal transport theory and Riemannian geometry. Ricci curvature and entropy are the key ingredients. See [Lo2] for a survey in the same spirit with a slightly different selection of topics.

Optimal transport theory is concerned with the behavior of transport between two probability measures in a metric space. We say that such transport is optimal if it minimizes a certain cost function typically defined from the distance of the metric space. Optimal transport naturally inherits the geometric structure of the underlying space; in particular Ricci curvature plays a crucial role for describing optimal transport in Riemannian manifolds. In fact, optimal transport is always performed along geodesics, and we obtain Jacobi fields as their variational vector fields. The behavior of these Jacobi fields is controlled by the Ricci curvature as is usual in comparison geometry. In this way, a lower Ricci curvature bound turns out to be equivalent to a certain convexity property of entropy in the space of probability measures.

Introduction

Modern medical imaging modalities can provide a great amount of information to study the human anatomy and physiological functions in both space and time. In cardiac magnetic resonance imaging (MRI), for example, several slices can be acquired to cover the heart in 3D and at a collection of discrete time samples over the cardiac cycle. From these partial observations, the challenge is to extract the heart's dynamics from these input spatio-temporal data throughout the cardiac cycle [10, 12].

Image registration consists in estimating a transformation which insures the warping of one reference image onto another target image (supposed to present some similarity). Continuous transformations are privileged; the sequence of transformations during the estimation process is usually not much considered. Most important is the final resulting transformation, not the way one image will be transformed to the other. Here, we consider a reasonable registration process to continuously map the image intensity functions between two images in the context of cardiac motion estimation and modeling.

The aim of this chapter is to present, in the context of extended optical flow (EOF), an algorithm to compute a time-dependent transportation plan without using Lagrangian techniques.

This book contains the proceedings of the summer school “Optimal transportation: Theory and Applications” held at the Fourier Institute (University of Grenoble I, France). The first 2 weeks were devoted to courses that described the main properties of optimal transportation and discussed its applications to analysis, differential geometry, dynamical systems, partial differential equations and probability theory. Courses were addressed both to students and researchers. A workshop took place during the last week. The aim of this conference was to present very recent developments of optimal transportation and also its applications in biology, mathematical physics, game theory and financial mathematics.

The first part of the book contains (expanded) versions of the courses. There are two sets of notes by F. Santambrogio. The first one gives a short introduction to optimal transport theory. In particular, the Kantorovich duality, the structure of Wasserstein spaces and the Monge–Ampère equations related to optimal transport are presented to the readers. These notes could be seen as an introduction for the other papers of the book. The second one describes applications to economics, game theory and urban planning.

The notes of I. Gentil, P. Topping and S.-I. Ohta describe (with different flavours) the connections between optimal transport and the notion of Ricci curvature, which is a very important tool in classical Riemannian geometry. A notion of curvature-dimension condition was defined by D. Bakry and M. Émery to study geometric properties of diffusions and to get functional inequalities.

Introduction

This chapter, which is an accompanying paper to [BLS09], consists of two parts. In Section 9.2 we present a version of Fenchel's perturbation method for the duality theory of the Monge–Kantorovich problem of optimal transport. The treatment is elementary as we suppose that the spaces (X, μ), (Y, ν), on which the optimal transport problem [Vil03, Vil09] is defined, simply equal the finite set {1, …, N} equipped with uniform measure. In this setting the optimal transport problem reduces to a finite-dimensional linear programming problem.

The purpose of this first part of the paper is rather didactic: it should stress some features of the linear programming nature of the optimal transport problem, which carry over also to the case of general Polish spaces X, Y equipped with Borel probability measures μ, ν, and general Borel measurable cost functions c : X × Y → [0, ∞]. This general setting is analysed in detail in [BLS09]; Section 9.2 may serve as a motivation for the arguments in the proof of Theorems 1.2 and 1.7 of [BLS09] which pertain to the general duality theory.

The second – and longer – part of the paper, consisting of Sections 9.3 and 9.4, is of a quite different nature. Section 9.3 is devoted to illustrating a technical feature of [BLS09, Theorem 4.2] by an explicit example.

Introduction

The goal of this course (given in 2009 in Grenoble) is to introduce inequalities as Poincaré or logarithmic Sobolev for diffusion semigroups. We will focus more on examples than on the general theory. A main tool to obtain those inequalities is the so-called Bakry–Emery Γ2-criterion. This criterion is well known to prove such inequalities and has also been used many times for other problems; see, for instance, [BÉ85, Bak06]. We will focus on the example of the Ornstein–Uhlenbeck semigroup and on the Γ2-criterion.

In Section 3.2 we investigate the main example of the Ornstein–Uhlenbeck semigroup, whereas in Section 3.3 we show how the Γ2-criterion implies such inequalities. In Section 3.4 we will explain an alternative method to get a logarithmic Sobolev inequality under curvature assumption. It is called the mass transportation method and has been introduced recently; see [CE02, OV00, CENV04, Vil09]. In this way we will also obtain another inequality called the Talagrand inequality or T2inequality.

The Ornstein–Uhlenbeck semigroup and the Gaussian measure

In the general setting, if (Xt)t≥0 is a Markov process on ℝn, then the family of operators

Pt(f)(x) = E(f (Xt)),

where X0 = x and a smooth function f, is defined as a Markov semigroup on ℝn. There are two main examples. The first one is the heat semigroup, which is associated with the Brownian motion on ℝn.