  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.
