Preunkert, Marc A semigroup version of the isoperimetric inequality. (English) Zbl 1093.47042 Semigroup Forum 68, No. 2, 233-245 (2004). The isoperimetric inequality in \({\mathbb R}^n\) states the following principle. Let \(A\), \(B\) be subsets of \({\mathbb R}^n\) with the same volume, \(B\) a Euclidean ball and denote by \(| {\partial}A|\), \(| {\partial}B|\) the respective measure of the surfaces \({\partial}A\) and \({\partial}B\). Then the inequality \(| {\partial}A|\geq |{\partial}B|\) holds. In the present paper, the author develops a semigroup version of this inequality. For this purpose, the notion of perimeter of a Cacciopoli set (introduced by E. De Giorgi) yields the appropriate measure theoretical background. In the proofs, the author uses properties of the heat semigroup as well as its explicit integral representation. Reviewer: Neculai Papaghiuc (Iaşi) Cited in 14 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 49Q20 Variational problems in a geometric measure-theoretic setting Keywords:heat semigroup; isoperimetric inequality; Cacciopoli sets Citations:Zbl 0841.49024; Zbl 0051.29403; Zbl 0942.49503 PDFBibTeX XMLCite \textit{M. Preunkert}, Semigroup Forum 68, No. 2, 233--245 (2004; Zbl 1093.47042) Full Text: DOI