Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives (Cambridge Tracts in Mathematics, Series Number 145)

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 qualitiative features of the inequality that 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, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day - and beyond!

This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject.

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As part of a series whose volumes pursue a narrow path in a broad subject, this introductory tract follows two venues of the isoperimetric inequality which span the evolution from the Euclidian space of classical differential geometry to the inequalities of modern measure theory. The treatment of Riemann manifolds is guided by the dichotomy between their local Euclidian character and global geometric properties, and covers discretization techniques and application to large time heat diffusion problems. Chapters include proofs. Chavel (mathematics, City U. of New York) is the author of and . Annotation c. Book News, Inc., Portland, OR (booknews.com)

This advanced introduction emphasises the variety of ideas, techniques, and applications of isoperimetric inequalities