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

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Let \(\psi : {\mathbb R}^n \longrightarrow {\mathbb R}^k\) be a map defined by \(k\) positive definite quadratic forms on \({\mathbb R}^n\). 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 \(a=\left( a_1, \ldots , a_k\right) \) in the convex hull of the image of \(\psi \) satisfying \(a_1 + \cdots + a_k=1\) there is a point \(b=\left( b_1, \ldots , b_k\right) \) in the image of \(\psi \) satisfying \(b_1 + \cdots + b_k =1\) and such that \(\sum _{i=1}^k a_i \ln \left( a_i/b_i\right) < 4.8\). 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 \(\sum _{i=1}^k a_i \ln \left( a_i/b_i\right) < 17/\sqrt{m}\).