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A few remarks on isoperimetric inequalities. (English) Zbl 0819.53002

Summary: This survey presents different topics from the vast area of isoperimetric problems. It is based on the huge literature dealing with it, in particular on the very fine books by C. Bandle, Bonnesen, Burago- Zalgaller, Fejes Tóth and Pólya-Szegő and on the survey articles by Bandle, Osserman and Payne. From these sources the two excellent papers [Bull. Am. Math. Soc. 84, 1182-1238 (1978; Zbl 0411.52006) and Am. Math. Mon. 86, 1-29 (1979; Zbl 0404.52012)] by R. Osserman deserve particular praise: They present in a very clear, highly readable and well organized form central parts of the topic isoperimetry. Therefore heavy use has been made of them. The presentation is elementary and every effort has been made to make it easy reading. Proofs are missing in most cases. The bibliography is quite detailed so that a reader can find his way to original papers in case he desires.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
51M25 Length, area and volume in real or complex geometry
52A40 Inequalities and extremum problems involving convexity in convex geometry
53C40 Global submanifolds
53A05 Surfaces in Euclidean and related spaces
53A15 Affine differential geometry
53C65 Integral geometry
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