Some Isoperimetric Inequalities in the Plane with Radial Power Weights.

We consider the punctured plane with volume density | x | α and perimeter density | x | β . We show that centred balls are uniquely isoperimetric for indices (α , β) which satisfy the conditions α - β + 1 > 0 , α ≤ 2 β and α (β + 1) ≤ β 2 except in the case α = β = 0 which corresponds to the classical isoperimetric inequality. As an application, we verify a conjecture due to Caldiroli and Musina relating to the best constant in the Caffarelli–Kohn–Nirenberg inequality. [ABSTRACT FROM AUTHOR]