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The standard isoperimetric theorem. (English) Zbl 0799.51015
Gruber, P. M. (ed.) et al., Handbook of convex geometry. Volume A. Amsterdam: North-Holland. 73-123 (1993).
This article as a chapter of the ‘Handbook of convex geometry’ by P. M. Gruber and J. M. Wills (1993; Zbl 0777.52001) is a report on the standard isoperimetric theorem. After historical remarks the author presents the ideas in this field from an analytic standpoint. There are many important references. From author’s summary: “A highly informative prototypal theorem says that among all subsets of the Euclidean \(n\)- dimensional space \(\mathbb{R}^ n\) having a prescribed measure the balls take the smallest perimeter. The present chapter is devoted to a thorough treatment of this theorem. The proof offered here stems from direct methods of the calculus of variations, and consists of two main steps. (i) A set which renders the perimeter a minimum does actually exist in the class of all subsets of \(\mathbb{R}^ n\) that have a prescribed measure and lie within a given large ball. (ii) Any relevant minimizer is virtually a ball. Step (i) – existence of minimizers – is a matter of building up an appropriate perimeter, i.e., a set function which continues the elementary perimeter, moreover makes semicontinuity and compactness arguments work. Step (ii) – identification of minimizers – involves Steiner symmetrizations. The theory of \(BV (\mathbb{R}^ n)\), a space of functions whose derivatives are measures, and the parallel theory of Caccioppoli sets, the subsets of \(\mathbb{R}^ n\) whose characteristic functions are in \(BV (\mathbb{R}^ n)\), provide the appropriate framework. For the sake of completeness, and alternative proof, due to Hurwitz and Lebesgue, is given in case the dimension \(n\) is 2”.
For the entire collection see [Zbl 0777.52001].
Reviewer: J.Böhm (Jena)

51M16 Inequalities and extremum problems in real or complex geometry
51M25 Length, area and volume in real or complex geometry
26B15 Integration of real functions of several variables: length, area, volume
28A75 Length, area, volume, other geometric measure theory
49Q15 Geometric measure and integration theory, integral and normal currents in optimization