  Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys.
Rev. Lett., 2004), we consider a variational plate model for an elastic cone
under compression in the direction of the cone symmetry axis. Assuming radial
symmetry, and modeling the compression by suitable Dirichlet boundary
conditions at the center and the boundary of the sheet, we identify the energy
scaling law in the von-K\'arm\'an plate model. Specifically, we find that three
different regimes arise with increasing indentation $\delta$: initially the
energetic cost of the logarithmic singularity dominates, then there is a linear
response corresponding to a moderate deformation close to the boundary of the
cone, and for larger $\delta$ a localized inversion takes place in the central
region. Then we show that for large enough indentations minimizers of the
elastic energy cannot be radially symmetric. We do so by an explicit
construction that achieves lower elastic energy than the minimum amount
possible for radially symmetric deformations.
