Description
This text discusses the classical isoperimetric inequality in Euclidean space and rough inequalities in noncompact Riemannian manifolds. It focuses on the most general form of the inequality and its applications in both classical differential geometry and modern geometric measure theory. The book also covers discretization techniques and applications to heat diffusion in Riemannian manifolds. It is praised for its clear and elegant presentation and its usefulness for graduate students and researchers in analysis and geometry.
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. Review: 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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