Geometric Measures And Affine Isoperimetric Type Problems

This dissertation is devoted to the Brunn-Minkowski theory,and deals with some geometric measures and affine isoperimetric type problems.The involved topics are John ellipsoids,cone volume measures,cone volume functionals,mixed volume and dual affine quermassintegrals.In Chapter 2,a class of extremum problems on L0 mixed volume are solved.Con-sequently,a class of associated ellipsoids,called logarithmic John ellipsoid,are estab_lished for convex bodies.And the essential connection between the characterization of logarithmic John ellipsoid and the isotropy of cone volume measure is demonstrated.The volume-ratios of logarithmic John ellipsoid are studied.Especially,a volume-ratio inequality is established.Convex bodies with identical John and logarithmic ellipsoids are characterized.In Chapter 3,a new geometric measure called mixed cone volume measure,and a new geometric quantity called mixed cone volume functional are introduced.And the affine invariance of mixed cone volume measure and mixed cone volume functional are proved.Several affine inequalities are established for volume,mixed volume and mixed cone volume functional.In Chapter 4,a new geometric quantity called quasi-Lp mixed volume is introduced by mixed cone volume measure.And a class of extremum problems on quasi-Lp mixed volume are solved.Consequently,a class of associated ellipsoids,called mixed Lp John ellipsoids,are established for convex bodies.The inclusion and volume-ratios of mixed Lp John ellipsoids are studied.Especially,both inclusion and volume-ratio inequality are established.In Chapter 5,the first harmonic variations of dual affine quermassintegrals are s-tudied.A class of geometric quantities,called mixed dual affine quermassintegrals,are introduced.And the affine invariance of mixed dual affine quermassintegrals are proved.A class of extremum problems on mixed dual affine quermassintegrals are solved.Con-sequently,a class of associated ellipsoids,called intersection mean ellipsoids,are estab-lished for convex bodies.The continuity and the affine invariance of intersection mean ellipsoids are proved.And several affine isoperimetric inequalities are established.