  We introduce an evolution model \`a la Firey for a convex stone which tumbles
on a beach and undertakes an erosion process depending on some variational
energy, such as torsional rigidity, principal Dirichlet Laplacian eigenvalue,
or Newtonian capacity. Relying on the assumption of existence of a solution to
the corresponding parabolic flow, we prove that the stone tends to become
asymptotically spherical. Indeed, we identify an ultimate shape of these flows
with a smooth convex body whose ground state satisfies an additional boundary
condition, and we prove symmetry results for the corresponding overdetermined
elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies:
we introduce new notions of cone variational measures and we prove that, if
such a measure is absolutely continuous with constant density, the underlying
body is a ball.
