  A developable cone ("d-cone") is the shape made by an elastic sheet when it
is pressed at its center into a hollow cylinder by a distance $\epsilon$.
Starting from a nonlinear model depending on the thickness $h > 0$ of the
sheet, we prove a $\Gamma$-convergence result as $h \rightarrow 0$ to a
fourth-order obstacle problem for curves in $\mathbb{S}^2$. We then describe
the exact shape of minimizers of the limit problem when $\epsilon$ is small. In
particular, we rigorously justify previous results in the physics literature.
