# Convex sets

Saturday, January 20, 2007 - 2:30pm - 3:20pm

Jean Lasserre (Centre National de la Recherche Scientifique (CNRS))

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

K and any linear polynomial f, is a sum of squares. We then

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 withK and any linear polynomial f, is a sum of squares. We then

Friday, January 19, 2007 - 2:30pm - 3:20pm

Victor Vinnikov (Ben Gurion University of the Negev)

I will discuss the characterization of convex sets in

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

^{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