A note on the convex body isoperimetric conjecture in the plane

TL;DR

This paper confirms two specific cases of the convex body isoperimetric conjecture in the plane, showing that certain symmetric and near-circular convex sets minimize perimeter for a given area.

Contribution

It verifies the conjecture for convex sets symmetric to coordinate axes and for perturbations of the unit disk, advancing understanding of the conjecture's validity.

Findings

Confirmed the conjecture for symmetric convex sets

Validated the conjecture for perturbations of the unit disk

Provided partial progress on the general conjecture

Abstract

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture: domains symmetric to both coordinate axes and perturbations of unit disk.