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• W2497378839 abstract "A highly informative prototypal theorem says that among all subsets of Euclidean n-dimensional space ℝn 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 Rn 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 - the one fitting polyhedra and nice subsets of ℝn - moreover makes semicontinuity and compactness arguments work. Step (ii) - identification of minimizers - involves Steiner symmetrizations. The theory of BV(ℝn), a space of functions whose derivatives are measures, and the parallel theory of Caccioppoli sets, the subsets of ℝn whose characteristic functions are in BV(ℝn), provide the appropriate framework. For the sake of completeness, an alternative proof, due to Hurwitz and Lebesgue, is given in case the dimension n is 2." @default.
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• W2497378839 creator A5069544328 @default.
• W2497378839 date "1993-01-01" @default.
• W2497378839 modified "2023-10-12" @default.
• W2497378839 title "The Standard Isoperimetric Theorem" @default.
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• W2497378839 doi "https://doi.org/10.1016/b978-0-444-89596-7.50008-0" @default.
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