  Convex algebraic geometry concerns the interplay between optimization theory
and real algebraic geometry. Its objects of study include convex semialgebraic
sets that arise in semidefinite programming and from sums of squares. This
article compares three notions of duality that are relevant in these contexts:
duality of convex bodies, duality of projective varieties, and the
Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the
optimal value of a polynomial program is an algebraic function whose minimal
polynomial is expressed by the hypersurface projectively dual to the constraint
set. We give an exposition of recent results on the boundary structure of the
convex hull of a compact variety, we contrast this to Lasserre's representation
as a spectrahedral shadow, and we explore the geometric underpinnings of
semidefinite programming duality.
