  Let psi: R^n --> R^k be a map defined by k positive definite quadratic forms
on R^n. We prove that the relative entropy (Kullback-Leibler) distance from the
convex hull of the image of psi to the image of psi is bounded above by an
absolute constant. More precisely, we prove that for every point a=(a_1, ...,
a_k) in the convex hull of the image of psi such that a_1 + ... +a_k =1 there
is a point b=(b_1, ..., b_k) in the image of psi such that b_1 + ... + b_k =1
and such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 4.8. Similarly, we
prove that for any integer m one can choose a convex combination b of at most m
points from the image of psi such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k)
< 15/sqrt{m}.
