  We prove bounds for the number of solutions to
  $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals,
which are sufficiently convex or near-convex. A near-convex set will be the
image of a set with small additive doubling under a convex function with
sufficiently many strictly monotone derivatives. We show, roughly, that every
time the number of terms in the equation is doubled, an additional saving of
$1$ in the exponent of the trivial bound $N^{2k-1}$ is made, starting from the
trivial case $k=1$. In the context of near-convex sets we also provide explicit
dependencies on the additive doubling parameters.
  Higher convexity is necessary for such bounds to hold, as evinced by sets of
perfect powers of consecutive integers. We exploit these stronger assumptions
using an idea of Garaev, rather than the ubiquitous Szemer\'edi-Trotter
theorem, which has not been adapted in earlier results to embrace higher
convexity.
  As an application we prove small improvements for the best known bounds for
sumsets of convex sets under additional convexity assumptions.
