Convex sets

We provide a sufficient condition on a class of
compact basic semialgebraic sets K for their convex hull
to have a lifted semidefinite representation (SDr). This lifted
SDr
is explicitly expressed in terms of the polynomials that define
K.
Examples are provided. For convex and compact basic
semi-algebraic sets
K defined by concave polynomials,
we also provide an explicit lifted SDr when the nonnegative
Lagrangian
L_f associated with
K and any linear polynomial f, is a sum of squares. We then

I will discuss the characterization of convex sets in ^m
which can be represented by Linear Matrix Inequalities, i.e., as feasible
sets of semidefinite programmes. There is a simple necessary condition,
called rigid convexity, which has been shown to be sufficient for sets in
the plane and is conjectured to be sufficient (in a somewhat weakened
sense) for any m.

This should be contrasted with the situation for matrix convex sets that