基于相对熵距离的二次映射象的凸性
Convexity of the image of a quadratic map via the relative entropy distance
摘要：
设ψ：R^n --> R^k是R^n上由k个正定二次型定义的映射.我们证明了从psi像的凸包到psi像的相对熵（Kullback-Leibler）距离上有界于一个绝对常数。更准确地说，我们证明了对于每个点a=（a_1，...，a_k）在psi的图像的凸包中，使得a_1 +. +a_k =1存在点b=（b_1，.，b_k）在psi的图像中，使得b_1 +. + b_k =1且使得a_1ln（a_1/b_1）+. + a_k ln（a_k/b_k）< 4.8。同样，我们证明，对于任何整数m可以选择一个凸组合B的最多m点的图像psi，使a_1 ln（a_1/b_1）+. + a_k ln（a_k/b_k）< 15/sqrt{m}。
英文摘要：
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}.
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