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Chapter 2 · Book pp. 48–95 · August 9, 2026 edition

Functional Inequalities

Develop Poincaré, log-Sobolev, transport, and isoperimetric tools, including semigroup proofs and preservation operations.

Begin with 2.1 Open this chapter in the canonical August 9 source ↗
Formal topologyOpen this chapter's Lean branches↗

Chapter route

This chapter develops Poincaré inequality, log-Sobolev inequality, transport inequalities, Cheeger inequality. Its main destination is to connect the definitions below to the results that later chapters consume.

Core definitions

  • Poincaré inequality: variance is controlled by Dirichlet energy; the compiled local interface records probability normalization, its test class, and all three integrability requirements explicitly.
  • Log-Sobolev inequality: relative entropy is controlled by Fisher information.
  • Talagrand transport inequalities compare relative entropy with Wasserstein distance.
  • Isoperimetric and concentration profiles quantify boundary expansion and tail decay.

Main results

  • Markov-semigroup interpolation proves functional inequalities from curvature and dissipation.
  • Tensorization, bounded perturbation, Lipschitz pushforward, and other operations preserve selected inequalities with tracked constants.
  • Functional inequalities imply concentration and isoperimetric estimates.
  • The framework extends, with changed analytic interfaces, to manifolds and discrete chains.

Contents

  1. 2.1Overview of the InequalitiesBook p. 48
  2. 2.2Proofs via Markov Semigroup TheoryBook p. 50
  3. 2.3Operations Preserving Functional InequalitiesBook p. 60
  4. 2.4Concentration of Measure and IsoperimetryBook p. 68
  5. 2.5Riemannian ManifoldsBook p. 77
  6. 2.6Discrete Space and TimeBook p. 84
  7. 2.bibBibliographical NotesBook p. 85
  8. 2.exExercisesBook p. 87
Why is this chapter route valid?

Analytic contracts

  • State whether Hessian inequalities hold everywhere, almost everywhere, or in a weak convex-analytic sense.
  • Track normalization and absolute continuity whenever a potential is used to define a probability law.
  • Keep localization inputs separate from the one-dimensional inequality they reduce to.

Open boundaries

  • Bakry–Émery Poincaré criterion, pending a concrete semigroup/generator domain
  • Full localization theorem
  • Dimension-sharp log-concave isoperimetry
  • Complete perturbation hierarchy
View Lean formalization

These mappings are evidence links, not a claim that the entire chapter is formalized.

Official source supplement

Supplement to Chapter 2

Open Chewi's supp.pdf ↗

The supplement says that this material was omitted from Log-Concave Sampling for space. Samplinglib therefore maps all sections of supp.pdf into the Chapter 2 source trail. The cards below summarize and anchor the source rather than republishing the 20-page document wholesale.

S2.1 · supp. §1.1 · pp. 2-4

Proof of Marton's tensorization

Marton's product-space tensorization of transport inequalities, including the conditional-coupling induction and KL chain rule.

  • Theorem 1.1: Marton's tensorization
  • Example 1.1: tensorization of T1
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S2.2 · supp. §1.2 · pp. 4-9

Concentration of measure

Mean/median equivalence, Orlicz norms, the Herbst argument, and transport-inequality characterizations of concentration.

  • §1.2.1 Equivalence between the mean and the median
  • §1.2.2 The Herbst argument
  • §1.2.3 Transport inequalities and concentration
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S2.3 · supp. §1.3 · pp. 9-11

Tensorization and Gozlan's theorem

Dimension-free concentration through tensorization and the converse route from product-space concentration to T2.

  • Empirical-measure Wasserstein estimate
  • Sanov theorem input
  • Gozlan's equivalence for T2
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S2.4 · supp. §1.4 · pp. 11-17

Metric measure spaces

Metric geometry, Alexandrov curvature, Lott-Sturm-Villani synthetic Ricci curvature, measured Gromov-Hausdorff stability, and comparison with the Bakry-Emery viewpoint.

  • §1.4.1 Metric geometry
  • §1.4.2 Lott-Sturm-Villani synthetic Ricci curvature
  • §1.4.3 Discussion
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S2.5 · supp. §1.5 · pp. 17-19

Exercises

Exercises fill additional proof details and connect the supplement to standard concentration inequalities.

  • Exercises 1.1-1.5
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S2.6 · supp. §References · pp. 19-20

References

Primary literature and standard geometry/optimal-transport references supporting the omitted Chapter 2 material.

  • Ambrosio-Gigli-Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures
  • Burago-Burago-Ivanov, A Course in Metric Geometry
  • Bobkov-Gotze on transport and logarithmic Sobolev inequalities
  • Lott-Villani and Sturm on synthetic Ricci curvature
  • Marton on transport tensorization
  • Villani, Optimal Transport: Old and New
Open exact source pages ↗