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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 theimage of a set with small additive doubling under a convex function withsufficiently many strictly monotone derivatives. We show, roughly, that everytime 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 thetrivial case $k=1$. In the context of near-convex sets we also provide explicitdependencies on the additive doubling parameters.  Higher convexity is necessary for such bounds to hold, as evinced by sets ofperfect powers of consecutive integers. We exploit these stronger assumptionsusing an idea of Garaev, rather than the ubiquitous Szemer\'edi-Trottertheorem, which has not been adapted in earlier results to embrace higherconvexity.  As an application we prove small improvements for the best known bounds forsumsets of convex sets under additional convexity assumptions.

Higher Convexity and Iterated Second Moment Estimates