Preprint

Inserted: 16 oct 2026

Year: 2026
Doi: https://doi.org/10.48550/arXiv.2610.15088

Abstract:

Among connected, planar sets, the ratio between the isoperimetric deficit and the squared barycentric asymmetry is known to be bounded from below by a strictly positive constant (as proved in recent papers by Bianchini-Croce-Henrot and Gambicchia-Pratelli). This opens the natural question whether the infimum is attained at some set. In this paper, we prove that a minimizer for this ratio actually exists, and moreover it is simply connected.

Keywords: dumbbell conjecture, isoperimetric inequality, barycentric ratio, asymmetry