Convexity in SemiAlgebraic Geometry and Polynomial Optimization

We review several (and provide new) results on the theory of moments, sums of squares, and basic semialgebraic 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 semialgebraic set $\mathbf{K}$ is convex but its defining polynomials are not, we provide two algebraic certificates of convexity which can be checked numerically. The second is simpler and holds if a sufficient (and almost necessary) condition is satisfied; it also provides a new condition for $\mathbf{K}$ to have semidefinite representation. For this we use (and extend) some of the recent results from the author and Helton and Nie [Math. Program., to appear]. 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.