凸函数Steiner对称化的一个等价特征

函数Steiner对称化的经典定义是根据函数水平集的Steiner对称化以及函数的分层表示定义的.给出了强制凸函数Steiner对称化的一个解析表达式,它是经典Steiner对称化的一个等价特征.这个新的定义不依赖于函数水平集的Steiner对称化,而是将定义转化为一维的类似抛物线函数的Steiner对称化,这更有助于函数不等式的证明.函数的Blaschke-Santalo不等式是一个重要的函数形式的仿射等周不等式,它的几何背景是凸体的BlaschkeSantalo不等式.首先利用Steiner对称化的新的定义,证明了对于任意的强制凸函数经过一次Steiner对称化后积分值变小了;然后利用Prekopa-Leindler不等式证明了径向函数的Blaschke-Santalo不等式.由于任何凸函数经过不断的Steiner对称化总可以在Lp范数意义下收敛于它的对称递减重排,而对称递减重排即为径向函数,因此证明了函数形式的Blaschke-Santalo不等式.

An Equivalent Characterization of the Steiner Symmetrization of Convex Functions

The classical definition of the functional Steiner symmetrization is defined according to the Steiner symmetrization of level sets of the function and the layer cake representation. In this paper, we give an analytic expression for the Steiner symmetrization of coercive convex functions, which is an equivalent characterization of the classical Steiner symmetrization. This new definition does not depend on the Steiner symmetrization of the level sets; instead, it converts the definition into the symmmetrization of a one-dimensional parabolic function, which is more helpful to prove the functional inequality. The functional Blaschke-Santalo inequality is an important functional affine isoperimetric inequality. Its geometric background is the Blaschke-Santalo inequality of convex bodies. In this paper, using the new definition, we first prove that the integral value of the convex function is reduced with respect to Steiner symmetrization and then, using Prekopa-Leindler inequality, we prove the Blaschke-Santalo inequality of radial function. By continuous Steiner symmetrizations, a convex function can always be converged to its symmetric decreasing rearrangement in the sense of Lp norm, and the symmetric decreasing rearrangement is a radial function, so the functional Blaschke-Santalo inequality is proved.