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The isoperimetric theorem for general integrands. (English) Zbl 0923.49019
§1 contains historical and bibliographical notes about isoperimetric problems. §2 refines the terminology for sets with finite perimeter in n , n2. §3 defines a positive, constant coefficient, parametric integrand of degree n-1 on n . §4 gives a new proof for the following isoperimetric theorem: “Let ψ be a norm on n , or, more generally, a positive, constant coefficient, parametric integrand of degree n-1 (not necessarily even or convex). Let B ψ ={x:ψ * (x)1} be the unit ball or the Wulff crystal. Let Ω be any measurable subset of n with finite perimeter and of the same volume as B ψ . Then ψ(Ω)ψ(B ψ ), with equality holding if and only if Ω differs from a translate of Bψ by a set of volume zero.” §5 derives an extension of the Gauss-Green-Federer divergence theorem which is suitable for the purposes of this paper.
MSC:
49Q20Variational problems in a geometric measure-theoretic setting
52A40Inequalities and extremum problems (convex geometry)