关于对数凹函数的随机单纯形体积不等式

The Volume Inequality of Random Simplices of Log-Concave Functions

摘要: 过去，关于凸体的随机单纯形体积不等式已经被证明，当1 ≤ k < n时，等号成立的充要条件是凸体K是单位球；k = n时，等号成立的充要条件是凸体K是中心在原点的椭球。该文证明了对数凹函数的随机单纯形体积不等式，当1 ≤ k < n时，等号成立的充要条件是函数f与它的对称递减重排f^∗相等；当k = n时，等号成立的充要条件是f(x)=f^∗(Ax)，A∈SL(n)。

Abstract: In the past, the volume inequality of random simplex on convex bodies has been proved, when 1 ≤ k < n, the necessary and sufficient conditions for the equality sign to hold are that the convex body K is the unit sphere; when k = n, the necessary and sufficient conditions for the equality sign to hold are that the convex body K is an ellipsoid with its center at the origin. In this paper, we prove the volume inequality of random simplices of log-concave functions, when 1 ≤ k < n, the necessary and sufficient conditions for the equality sign to hold are that the function f is equal to its symmetric decreasing rearrangement f^∗; when k = n, the necessary and sufficient conditions for the equality sign to hold are that f(x)=f^∗(Ax), A∈SL(n).