The improved isoperimetric inequality and the Wigner caustic of planar ovals

TL;DR

This paper introduces an improved isoperimetric inequality for convex curves in the plane, incorporating the Wigner caustic's area, and establishes stability results showing near equality implies the curve is close to constant width.

Contribution

The paper presents a new inequality involving the Wigner caustic and proves stability, extending classical isoperimetric results with affine geometric insights.

Findings

Improved inequality includes Wigner caustic area term

Equality holds for curves of constant width

Stability result links near equality to curves close to constant width

Abstract

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4\pi A_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which states that if $M$ is a closed regular simple convex curve, then \begin{align*} L_{M}^2\geqslant 4\pi A_{M}+8\pi\left|\widetilde{A}_{E_{\frac{1}{2}}(M)}\right|, \end{align*} where $\widetilde{A}_{E_{\frac{1}{2}}(M)}$ is an oriented area of the Wigner caustic of $M$, and the equality holds if and only if $M$ is a curve of constant width. Furthermore we also present a stability property of the improved isoperimetric inequality (near equality implies curve nearly of constant width). The Wigner caustic is an example of an affine $\lambda$-equidistant (for $\displaystyle\lambda=\frac{1}{2}$) and the improved isoperimetric inequality is a consequence of certain bounds of oriented areas of affine equidistants.