| In convex geometry analysis and related fields,Steiner symmetrization is a very typical and useful tool in convex body theory.Although Jakob Steiner proposed the original intention of Steiner symmetry to solve the problem of equal week inequality,its function was immediately extended to other fields,such as the proof of some theorems of classical geometry and analysis requires the tool of Steiner symmetry.In this thesis,we first introduce the relevant convex geometric analysis concepts,such as level set,below and so on and I briefly reviewed the definition of Steiner symmetrization of functions and related properties.We know that the proof of low dimensional inequalities can use calculus and some simple techniques,but once the problem is extended to high dimension,there will often be some processing difficulties,then Steiner symmetrization will play a big role.On the basis of previous studies,I extend the space to the high-dimensional space,the following two problems are explored and solved:1.The Steiner inequality of the layering function in the log-concave function space.2.We generalize the log-concave function space to Sobolev space and prove the Steiner inequality for the hierarchical function in Sobolev space.This thesis concludes with a number of other applications of Steiner symmetrization,this can also see the value of Steiner symmetry. |
