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Look Inside Isoperimetric Inequalities

Isoperimetric Inequalities
Differential Geometric and Analytic Perspectives


Part of Cambridge Tracts in Mathematics

  • Date Published: July 2011
  • availability: Available
  • format: Paperback
  • isbn: 9781107402270

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About the Authors
  • This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitative features of the inequality which provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities.

    • Rich collection of ideas and techniques, from elementary to advanced
    • Transition from classical differential geometry to modern geometric measure theory
    • Was the first discussion of isoperimetric inequalities and large time heat diffusion in Riemannian manifolds in book form
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    Reviews & endorsements

    Review of the hardback: 'The presentation of the book is clear and elegant, and gives expression to the beauty of the subject. It is a great pleasure to read this book, which is a profound source text for both classical and modern methods, and which will be equally valuable to graduate students and researchers in analysis and geometry.' Bulletin of the London Mathematical Society

    Review of the hardback: 'The book is very useful in two ways. First, it nicely explains the story of the classical isoperimetric inequality, a result with a big disproportion between the ease of formulation and difficulty of the proof. This second part contains deep results obtained by the author.' European Mathematical Society

    Review of the hardback: '… very useful …' EMS Newsletter

    Review of the hardback: '[This book] constitues a valuable addition to the modern theory of inequalities.' Bulletin of the Belgian Mathematical Society

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    Product details

    • Date Published: July 2011
    • format: Paperback
    • isbn: 9781107402270
    • length: 282 pages
    • dimensions: 229 x 152 x 16 mm
    • weight: 0.42kg
    • availability: Available
  • Table of Contents

    Part I. Introduction:
    1. The isoperimetric problem
    2. The isoperimetric inequality in the plane
    3. Preliminaries
    4. Bibliographic notes
    Part II. Differential Geometric Methods:
    1. The C2 uniqueness theory
    2. The C1 isoperimetric inequality
    3. Bibliographic notes
    Part III. Minkowski Area and Perimeter:
    1. The Hausdorff metric on compacta
    2. Minkowski area and Steiner symmetrization
    3. Application: the Faber-Krahn inequality
    4. Perimeter
    5. Bibliographic notes
    Part IV. Hausdorff Measure and Perimeter:
    1. Hausdorff measure
    2. The area formula for Lipschitz maps
    3. Bibliographic notes
    Part V. Isoperimetric Constants:
    1. Riemannian geometric preliminaries
    2. Isoperimetric constants
    3. Discretizations and isoperimetric inequalities
    4. Bibliographic notes
    Part VI. Analytic Isoperimetric Inequalities:
    1. L2-Sobolev inequalities
    2. The compact case
    3. Faber-Kahn inequalities
    4. The Federer-Fleming theorem: the discrete case
    5. Sobolev inequalities and discretizations
    6. Bibliographic notes
    Part VII. Laplace and Heat Operators:
    1. Self-adjoint operators and their semigroups
    2. The Laplacian
    3. The heat equation and its kernels
    4. The action of the heat semigroup
    5. Simplest examples
    6. Bibliographic notes
    Part VIII. Large-Time Heat Diffusion:
    1. The main problem
    2. The Nash approach
    3. The Varopoulos approach
    4. Coulhon's modified Sobolev inequality
    5. The denoument: geometric applications
    6. Epilogue: the Faber–Kahn method
    7. Bibliographic notes

  • Author

    Isaac Chavel, City University of New York

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