Isoperimetric problem for exponential measure on the plane with $$\ell _1$$ ℓ 1 -metric

Abstract

We give a solution to the isoperimetric problem for the exponential measure on the plane with the 1 -metric. As it turns out, among all sets of a given measure, the simplex or its complement (i.e. the ball in the 1 -metric or its complement) has the smallest boundary measure. The proof is based on a symmetrisation (along the sections of equal 1 -distance from the origin).