Elasticae and inradius

TL;DR

This paper investigates the minimization of elastic energy for convex bodies with a fixed inradius, revealing that unlike other constraints, the optimal shape is not a disk and can be explicitly characterized.

Contribution

It introduces a novel minimization problem with inradius constraint and provides a complete elementary characterization of its solution, contrasting with previous results.

Findings

The minimizer is not a disk under inradius constraint.

Explicit elementary functions describe the optimal shape.

Contrasts with known solutions for other geometric constraints.

Abstract

The elastic energy of a planar convex body is defined by $E(\Om)=\frac 12\,\int\_{\partial\Om} k^2(s)\,ds$where $k(s)$ is the curvature of the boundary. In this paper we are interested in the minimization problemof $E(\Om)$ with a constraint on the inradius of $\Om$. By contrast with all the other minimization problemsinvolving this elastic energy (with a perimeter, area, diameter or circumradius constraints) for which thesolution is always the disk, we prove here that the solution of this minimization problem is not the disk and we completely characterizeit in terms of elementary functions.