Convexity of the image of a quadratic map via the relative entropy distance

Let (Formula presented.) be a map defined by k positive definite quadratic forms on Rn. We prove that the relative entropy (Kullback-Leibler) distance from the convex hull of the image of ψ to the image of ψ is bounded above by an absolute constant. More precisely, we prove that for every point (Formula presented.) in the convex hull of the image of ψ satisfying (Formula presented.) there is a point (Formula presented.) in the image of (Formula presented.) satisfying (Formula presented.) and such that (Formula presented.). Similarly, we prove that for any integer m one can choose a convex combination b of at most m points from the image of ψ such that (Formula presented.). © 2013, The Managing Editors.