Smoothness Of Steiner Symmetrizations

In 1986,Kiselman proved the smoothness of two-dimensional projection of three-dimensional smooth convex compact sets.On the basis of previous studies,we continue to further study the smoothness of Steiner symmetrization,and extend the smoothness of convex compact sets to convex functions and convex bodies.This paper mainly discusses the smoothness of Steiner symmetrization of C~1 convex bodies and C~2 convex functions,and studies and solves the following two problems:1.Based on the proved lemma that the Steiner symmetry of a C~1convex function is also a C~1convex function,the Steiner symmetry of a function in Sobolev space is extended to the Steiner symmetry of a convex body It is proved that the Steiner symmetry of a C~1convex body is still a C~1convex body.2.By extending the C~1convex function to the C~2convex function and further adding restrictions,it is proved that a C~2convex function with positive definite Hessian matrix is also a C~2convex function with positive definite Hessian matrix.This paper consists of five chapters.The first section introduces the background of this topic and the research status at home and abroad.The second chapter introduces some basic concepts and some basic property theorems needed in this paper.The third chapter introduces the proof of Steiner symmetry of C~1convex body.The fourth chapter introduces the proof of Steiner symmetry of C~2 convex function.The fifth chapter gives a summary of this paper and puts forward some directions for further research on Steiner symmetry smoothness.