| One of the most classical inequalities is isoperimetric inequality in mathematics,the geometric inequalities developed from it promote to the birth and development of several branches of mathematics.In Integral Geometry and Convex Geometric Analysis,the Aleksandrov-Fenchel inequality of quermassintegrals is a extension of classcial isoperimetric inequality and is one of the most core inequalities in mixed volume theory.The stronger conjecture is that the conjecture proposed by Lutwak about affine quermassintegrals,which contains the famous affine inequality Blaschke-Santalo inequality and Petty projection inequality.It was only recently that Milman-Yehudayoff completely solved this conjecture and generalized the affine quermassintegrals to L~p-moment quermassintegrals and obtained some inequalities related to quermassintegrals.In this paper,we first generalize the classical quermassintegrals and introduce the p-quermassintegrals in Chapter 3,we use the methods of Integral Geometry to obtain the isoperimetric inequality and the strengthened Uryson inequality of pquermassintegrals.We also define the projection entropy and get its isoperimetric inequality.Secondly,we generalize the L~p-moment quermassintegrals in Chapter 4,and define L~p-moment mixed quermassintegrals,and obtain the Brunn-Minkowski type inequality and the Aleksandrov-Fenchel type inequality which imply the Lutwak mixed polar projection inequality,etc.Finally,we discuss the radial mean bodies of the convex bodies.The radial mean body is proposed by Gardner-Zhang,which proved the opposite affine inequality of the Rogers-Shephard inequality and the Zhang projection inequality by the radial mean bodies.In Chapter 5,we show that the radial mean body of a planar triangle is a hexagon symmetric with respect to the origin,and give an exact counterexample to its inspecific monotonicity.we also discuss the inclusion relationship between its John ellipsoid and radial mean body. |
