The Affine Isocyclic Inequality On (p,q)-dual Curvature Measure

The isoperimetric problem and the Minkowski problem are core research contents of integral geometry and convex geometric analysis.The isoperimetric problem is equivalent to the isoperimetric inequality.The corresponding affine isoperimetric inequality is very important in integral geometry and convex ge-ometric analysis.To establish affine isoperimetric inequalities,affine invariants are the points.Geometric measures play an important role in the study of affine invariants.For example,the Lp surface area measure is the key to construct the Lp affine surface area,the Lp geominimal surface area,the Lp John ellipsoid.In 2018,Lutwak,Yang and Zhang posed a new family of geometric measures-Lp dual curvature measures(also called(p,q)-th dual curvature measures).It includes Lp surface area measures,Lp-integral curvatures and dual curvature measures and states the relation of the above three measures.The establishment of(p,q)-th dual curvature measures is a significant breakthrough in convex ge-ometry.Motivated by ideas of Lutwak,Yang and Zhang,the dissertation mainly discusses affine isoperimetric inequalities of the(p,q)-th dual curvature measures,that is,affine isoperimetric inequalities of the(p,q)-mixed affine surface area and the(p,q)-mixed geominimal surface area and geometric inequalities of the(p,q)-John ellipsoid.Chapter 3 firstly introduces the(p,q)-mixed geominimal surface area of the(p,q)-th dual curvature measures,which is an extension of the Lp geominimal sur-face area.The affine invariance under special linear transformations(Proposition 3.1.1)and the continuity(Theorem 3.1.2)of the(p,q)-mixed geominimal surface area are discussed.Because the definition of the(p,q)-mixed geominimal surface area is determined by an optimization problem,using the Blaschke selection the-orem we solve this optimization problem to give the concept of the(p,q)-mixed Petty bodies(Definition 3.1.3).The uniqueness(Theorem 3.1.1)and the uniform boundedness(Lemma 3.1.3)of the(p,q)-mixed Petty bodies are obtained.We also establish affine isoperimetric inequalities(Theorem 3.1.4)of the(p,q)-mixed geominimal surface area and the(p,q)-mixed Petty bodies paly an important role in equality conditionThen the(p,q)-mixed affine surface area of the(p,q)-th dual curvature mea-sures is investigated in Chapter 3.It is an extension of the Lp affine surface area Similarly the(p,q)-mixed affine surface area is affine invariant under special lin-ear transformations(Proposition 3.2.1)and is upper semicontinuous(Proposition 3.2.2).We also give the integral representation of the(p,q)-mixed affine surface area(Theorem 3.2.1).Affine isoperimetric inequalities of the(p,q)-mixed affine surface area(Theorem 3.2.3)are establishedFinally Chapter 4 discusses the(p,q)-John ellipsoids.The classical John el-lipsoid and the Lp John ellipsoid are special cases of the(p,q)-John ellipsoids.The(p,q)-John ellipsoids are characterized(Theorem 4.1.3).The continuity(Theo-rem 4.2.1)of the(p,q)-John ellipsoids is studied.Analogues of Ball’s volume-ratio inequality(Theorem 4.3.1)and John’s inclusion(Theorem 4.4.2)for the(p,q)-John ellipsoids are given.