Isoperimetric problem for exponential measure on the plane with l_1-metric

TL;DR

This paper solves the isoperimetric problem for the exponential measure on the plane with the l_1-metric, identifying simplices and their complements as the minimal boundary sets.

Contribution

It provides a solution to the isoperimetric problem in this specific geometric setting, using symmetrisation techniques.

Findings

Simplex and its complement minimize boundary measure for given measure

The minimal boundary sets are balls in the l_1-metric or their complements

Symmetrisation along sections of equal l_1-distance is key to the proof

Abstract

We give a solution to the isoperimetric problem for the exponential measure on the plane with the $\ell_1$-metric. As it turns out, among all sets of a given measure, the simplex or its complement (i.e. the ball in the $\ell_1$-metric or its complement) has the smallest boundary measure. The proof is based on a symmetrisation (along the sections of equal $\ell_1$-distance from the origin).