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arxiv: math/0602135 · v1 · submitted 2006-02-07 · 🧮 math.DG

On the isoperimetric problem in Euclidean space with density

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keywords densityisoperimetricconjectureeuclideanproblemproveregionsspace
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We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density $\exp (|x|^2)$ by using symmetrization techniques.

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