Note on the Isoperimetric Profile of a Convex Body

Ernst Kuwert ()
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Ernst Kuwert: der Albert-Ludwigs-Universität Freiburg, Mathematisches Institut

A chapter in Geometric Analysis and Nonlinear Partial Differential Equations, 2003, pp 195-200 from Springer

Abstract: Summary A solution to the relative isoperimetric problem (or partitioning problem) is a subset E of a given set Ω ⊂ ℝ n (the container) with prescribed volume |E| =V and minimal area $$ \int_\Omega {\left| {D_{XE} } \right|\, = \,A(V)} $$ of the interface. Improving a result due to Sternberg & Zumbrun [8], we obtain that if Ω is convex then $$ A{(V)^{{\frac{n}{{n - 1}}}}} $$ is a concave function of V. As consequence we deduce that the isoperimetric ratio of E is no worse than that of the half-ball contained in ℍ = ℝ n-1 X (0, ∞), and that the mean curvature of the minimizer is bounded a priori.

Keywords: Convex Body; Fundamental Form; Concave Function; Isoperimetric Inequality; Nonlinear Partial Differential Equation (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55627-2_12

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DOI: 10.1007/978-3-642-55627-2_12

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