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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)

##### MSC:
 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
##### Keywords:
isoperimetric problem; inequalities; extremum problem; perimeter