For 0 ≤ α < β+1 and β ≤ 2α, the weighted perimeter ∫ y^α ds is minimized among sets of fixed weighted measure ∬ y^β dx dy in R²₊ by an explicit y-axis symmetric set.
Proof of the Log-Convex Density Conjecture
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abstract
We completely characterize isoperimetric regions in R^n with density e^h, where h 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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2019 1verdicts
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Some isoperimetric inequalities with respect to monomial weights
For 0 ≤ α < β+1 and β ≤ 2α, the weighted perimeter ∫ y^α ds is minimized among sets of fixed weighted measure ∬ y^β dx dy in R²₊ by an explicit y-axis symmetric set.