  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.
