A weighted isoperimetric inequality in a wedge

TL;DR

This paper proves a weighted isoperimetric inequality in a wedge-shaped domain, showing that the intersection of the wedge with a centered ball minimizes the weighted perimeter for a fixed measure.

Contribution

It establishes a new isoperimetric inequality involving a specific weighted measure in a wedge, identifying the minimizers as centered ball intersections.

Findings

Centered ball intersections minimize weighted perimeter for fixed measure

The measure involves exponential and polynomial weights in the wedge

The result extends classical isoperimetric inequalities to weighted wedge domains

Abstract

Let $c, k_1,..., k_N $ be non-negative numbers, and define a measure $\mu $ in the wedge $W:= \{x\in \mathbb{R} ^N :\, x_i >0, i=1,...,N\} $ by $d\mu = e^{c|x|^2} x_1 ^{k_1}...x_N ^{k_N} \, dx $. It is shown that among all measurable subsets of $W$ with fixed $\mu$ -measure, the intersection of $W$ with a ball centered at the origin renders the weighted perimeter relative to $W$ a minimum.