Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance

TL;DR

This work analyzes the planar problem of minimizing $J(K)=\delta(K)/\lambda_0(K)^2$ for sets with fixed area $|K|=\pi$ under a diameter constraint $diam(K)\le D$. It proves the existence of minimizers for sufficiently large $D$, establishes global $C^{1,1}$ regularity with analytic boundary portions away from the barycentric region, and shows the optimal set consists of exactly two connected components touching the outer boundary at opposite points; notably, the boundary contains no arcs of circles. The authors derive a curvature-based optimality condition yielding a pendulum-type equation for the tangent angle, together with a detailed analysis of the roles of the barycentric and Fraenkel balls via shape derivatives. The results illuminate the geometry of optimal configurations and set the stage for further investigation into symmetry and finer structural descriptions.

Abstract

In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $δ(K)$ and the barycentric distance $λ_0(K)$ for sets $K\subset \mathbb{R}^N$ with given diameter $D$ and measure. In this work we are interested in the optimal sets for this inequality in the plane, i.e. sets that minimize the ratio $δ(K)/λ_0(K)^2$. We prove existence of optimal sets (at least when $D$ is large enough), regularity and express the optimality conditions. Moreover, we prove that the optimal sets have exactly two connected components and their boundary does not contain any arc of circle.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.1
• Theorem 3.1
• proof : Proof of Theorem \ref{['theoexistence']}