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Note on the Isoperimetric Profile of a Convex Body

In: Geometric Analysis and Nonlinear Partial Differential Equations

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  • Ernst Kuwert

    (der Albert-Ludwigs-Universität Freiburg, Mathematisches Institut)

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.

Suggested Citation

  • Ernst Kuwert, 2003. "Note on the Isoperimetric Profile of a Convex Body," Springer Books, in: Stefan Hildebrandt & Hermann Karcher (ed.), Geometric Analysis and Nonlinear Partial Differential Equations, pages 195-200, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55627-2_12
    DOI: 10.1007/978-3-642-55627-2_12
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