Elasticae and inradius

Abstract

The elastic energy of a convex curve bounding a planar convex body $\Omega$ is defined by $E(\Omega )=\frac{1}{2}\,\int _{\partial \Omega } k^2(s)\,ds$ where k(s) is the curvature of the boundary. In this paper we are interested in the minimization problem of $E(\Omega )$ with a constraint on the inradius of $\Omega$. In contrast to all the other minimization problems involving this elastic energy (with a perimeter, area, diameter, or circumradius constraints) for which the solution is always the disk, we prove here that the solution of this minimization problem is not the disk and we completely characterize it in terms of elementary functions.