Quantitative stability in the isodiametric inequality via the isoperimetric inequality

TL;DR

This paper links the isodiametric and isoperimetric inequalities through a variational principle, providing quantitative stability results and identifying nearly optimal sets that demonstrate sharpness.

Contribution

It introduces a variational approach connecting the inequalities, yielding new quantitative stability results and explicit nearly optimal sets.

Findings

Balls maximize perimeter among convex sets with fixed diameter

Quantitative improvements to the isodiametric inequality are sharp

Explicit nearly optimal sets demonstrate the bounds' sharpness

Abstract

The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle brings also quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets.