*preprint*

**Inserted:** 23 jul 2019

**Year:** 2019

**Abstract:**

We consider a thin elastic sheet in the shape of a sector that is clamped along the curved part of the boundary, and left free at the remainder. On the curved part, the boundary conditions agree with those of a conical deformation. We prove upper and lower bounds for the F\"oppl-von-K\'arm\'an energy under the assumption that the out-of-plane component of the deformation is convex. The lower bound is optimal in the sense that it matches the upper bound in the leading order with respect to the thickness of the sheet. As a corollary, we obtain a new estimate for the Monge-Amp\`ere equation in two dimensions.