Semidefinite representation for convex hulls of real algebraic curves

Understand

We show that the closed convex hull of any one-dimensional semi-algebraic subset of R^n has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality.

• This is proved by an application of the moment relaxation method.
• Given a nonsingular affine real algebraic curve C and a compact semialgebraic subset K of its R-points, the preordering P(K) of all regular functions on C that are nonnegative on K is known to be finitely generated.
• We prove that P(K) is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of P(K).