Convexity of the image of a quadratic map via the relative entropy distance (Q464812)

Let \(q_1, \dots, q_k :\mathbb{R}^n\to\mathbb{R}\) be positive definite quadratic forms and let \(\psi:\mathbb R^n\to\mathbb R^k\) be the corresponding map, \(\psi(x)=(q_1(x), \dots, q_k(x))\). Let \(a=(a_1, \dots, a_k)\) be a point in the convex hull of the image of \(\psi\) such that \(\sum^{k}_{i=1}a_i=1\). The author of the paper under review proves that there exists a point \(b= (b_1, \dots,b_k)\) in the image of \(\psi\) such that \(\sum^{k}_{i=1}b_i=1\) and such that \(\sum^{k}_{i=1}a_i \ln (a_i/ b_i)\leq \beta\), where \(\beta>0\) is an absolute constant. The author also proves that for any positive integer \(m\), there exists a point \(b= (b_1, \dots, b_k)\), such that \(\sum^{k}_{i=1}b_i=1\), the point \(b\) is a convex combination of at most \(m\) points of \(\psi(\mathbb{R}^n)\) and \(\sum^{k}_{i=1}a_i \ln (a_i/ b_i)<17/ \sqrt{m}\).