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

TL;DR

This paper studies the isoperimetric problem for the exponential measure on the plane under the $\ell_1$-metric and proves that among all Borel sets of a given measure, the simplex (or its complement) minimizes boundary measure, with the extremal choice determined by the radius $t$ so that $\nu(B_A)=\nu(A)$. The authors develop a symmetrisation along sections of equal $\ell_1$-distance from the origin and reduce the problem to trapezoids, ultimately establishing the inequality for $n=2$ and explaining partial results and limitations in higher dimensions. They also discuss the behavior for $1$-unconditional sets, noting that while balls minimize boundary in the unconditional class, balls are not extremal for the symmetric exponential measure on the plane. The results provide a precise isoperimetric structure for the plane with exponential measure and lay groundwork for related concentration phenomena, with conjectures for higher dimensions.

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).

Figures (3)

• Figure :
• Figure :
• Figure :

Theorems & Definitions (18)

• Definition 1
• Theorem 2
• Corollary 3
• Conjecture 4
• Lemma 5
• proof
• Corollary 6
• Lemma 7
• proof
• Lemma 8