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On the isoperimetric problem for radial logconvex densities
 Calc. Var. Partial Differential Equations
"... Abstract. Given a smooth, radial, uniformly logconvex density eV on Rn, n ≥ 2, we characterize isoperimetric sets E with respect to weighted perimeter R ∂E eV dHn−1 and weighted volume m = R E eV as balls centered at the origin, provided m ∈ [0, m0) for some (potentially computable) m0> 0; this ..."
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Abstract. Given a smooth, radial, uniformly logconvex density eV on Rn, n ≥ 2, we characterize isoperimetric sets E with respect to weighted perimeter R ∂E eV dHn−1 and weighted volume m = R E eV as balls centered at the origin, provided m ∈ [0, m0) for some (potentially computable) m0> 0; this affirmatively answers conjecture [RCBM, Conjecture 3.12] for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurcation values of the weighted volume parameter. Finally, we show the validity of the conjecture when V belongs to a C2neighborhood of cx2 (c> 0).
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"... Are circles isoperimetric in the plane with density er? ..."
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PROOF OF THE LOGCONVEX DENSITY CONJECTURE
"... Abstract. We completely characterize isoperimetric regions in Rn with density eh, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the LogConvex Density Conjecture due to Kenneth Brakke. ..."
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Abstract. We completely characterize isoperimetric regions in Rn with density eh, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the LogConvex Density Conjecture due to Kenneth Brakke.