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.
Cheeger, A lower bound for the smallest eigenvalue of the Laplacian,Problems in analysis: A symposium in honor of Salomon Bochner (1970), 195–199
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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.