  We review several (and provide new) results on the theory of moments, sums of
squares and basic semi-algebraic sets when convexity is present. In particular,
we show that under convexity, the hierarchy of semidefinite relaxations for
polynomial optimization simplifies and has finite convergence, a highly
desirable feature as convex problems are in principle easier to solve. In
addition, if a basic semi-algebraic set K is convex but its defining
polynomials are not, we provide a certificate of convexity if a sufficient (and
almost necessary) condition is satified. This condition can be checked
numerically and also provides a new condition for K to have semidefinite
representation. For this we use (and extend) some of recent results from the
author and Helton and Nie. Finally, we show that when restricting to a certain
class of convex polynomials, the celebrated Jensen's inequality in convex
analysis can be extended to linear functionals that are not necessarily
probability measures.
