Some isoperimetric inequalities in the plane with radial power weights

Abstract

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 ≤ 2β and α(β + 1) ≤ β 2 except in the case α = β = 0 which corresponds to the classical isoperimetric inequality.