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We completely characterize isoperimetric regions in with density , where is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.
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Gregory R. Chambers, Proof of the Log-Convex Density Conjecture. J. Eur. Math. Soc. 21 (2019), no. 8, pp. 2301–2332DOI 10.4171/JEMS/885