On a theorem of Schoen and Shkredov on sumsets of convex sets

Abstract: A set of reals $A={a_1,...,a_n}$ labeled in increasing order is called convex if there exists a continuous strictly convex function $f$ such that $f(i)=a_i$ for every $i$. Given a convex set $A$, we prove [|A+A|\gg\frac{|A|^{14/9}}{(\log|A|)^{2/9}}.] Sumsets of different summands and an application to a sum-product-type problem are also studied either as remarks or as theorems.