Research On The Convex Mixed Lorentz-Sobolev Inequality And Its Application

In this dissertation,isoperimetric problem and Minkowski problem are in-vestigated,which are the core of geometry.The isoperimetric inequality is de-rived from the isoperimetric problem,that is one of the oldest mathematical inequalities.The isoperimetric inequality has a profound affect on each branch in mathematics.The classical isoperimetric inequality is equivalent to the Sobolev inequality,it palys the role as a bridge which connect geometry and analysis.In 1999,Zhang established a new Sobolev inequality which is invariant under all affine transformations of R~nand turn out to be significantly stronger than the classical Sobolev inequality.Moreover,Zhang's affine Sobolev inequality is equivalent to the generalized Petty projection inequality.Inspire by Zhang's work,many more stronger affine Sobolev inequality have been found.Find analytic in-equalities which are equivalent to the known geometric inequalities is the central problem in convex geometric analysis.In 1993,the L_p-Minkowski mixed volume inequality for two convex bodies was obtained by Lutwak,that is one of the important inequality in convex geomet-ric analysis.In particular,if one convex body is the ball then the L_p-Minkowski mixed volume inequality precisely is the L_pisoperimetric inequality.The L_p isoperimetric inequality is the extension of the classical isoperimetric inequality,and it is equivalent to the known L_pSobolev inequality.In 2011,Ludwig-Xiao-Zhang obtained a sharp convex Lorentz-Sobolev inequality,and proved that the sharp convex Lorentz-Sobolev inequality is not only stronger than the known L_p Sobolev inequality but also equivalent to the L_pisoperimetric inequality.We first study the Orlicz-Minkowski problem(also called as the L_?-Minkowski problem,here?is a continuous function).In 2010,Haberl-Lutwak-Yang-Zhang obtained the existence of the Orlicz-Minkowski problem,when?is a decreasing continuous function.We solved the existence of the discrete Orlicz-Minkowski problem when?is an increasing function.Moreover,if?is a special function,our result is the discrete L_p(p<0)Minkowski problem that obtained by Zhu.Then we investigate the analysis inequality which is equivalent to the L_p-Minkowski mixed volume inequality.Minkowski problem is the key tool in the study of analysis inequality with geometric background.By the solution of L_p-Minkowski problem,we obtained a new Sobolev inequality.We proved that the new Sobolev inequality includes the L_pMinkowski mixed volume inequality.Our new Sobolev inequality implies the sharp convex Lorentz-Sobolev inequality obtained by Ludwig-Xiao-Zhang.Our new Sobolev inequality is called the sharp convex mixed Lorentz-Sobolev inequality.Inspired by Lutwak-Yang-Zhang's work on the L_pJohn ellipsoid,we defined the functional L_pJohn ellipsoid by using our sharp convex mixed Lorentz-Sobolev inequality.We also studied the mixed Pólya-Szeg? principle,and obtained some new mixed Sobolev inequalities.