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We solve a class of isoperimetric problems on RN with respect to weights that are powers of the distance to the origin. For instance we show that, if k∈[0,1], then among all smooth sets Ω in RN with fixed Lebesgue measure, ∫∂Ω|x|kHN−1(dx) achieves its minimum for a ball centered at the origin. Our results also imply a weighted Pólya–Szegö principle. In turn, we establish radiality of optimizers in...
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