Affine Extremum Problems In The Orlicz Brunn-Minkowski Theory

This dissertation is devoted to the Orlicz Brunn-Minkowski theory, and deals with some affine extremum problems and isoperimetric type inequalities. The involved topics are John ellipsoids, minimal surface area, quermassintegrals and affine quermassinte-grals.In Chapter 2, a class of affine extremum problems on Orlicz mixed volumes are solved. Consequently, a class of associated ellipsoids, called Orlicz-John ellipsoids, are established for convex bodies. The continuity of Orlicz-John ellipsoids and a com-mon limit position theorem are proved. It turns out that the Orlicz-John ellipsoids are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids in the framework of the Orlicz Brunn-Minkowski theory. The essential connection between the characterization of Orlicz-John ellipsoids and the isotropy of measures is demon-strated. The bounds and volume-ratios of Orlicz-John ellipsoids are studied. Especially, a volume-ratio inequality is established. This affine inequality generalizes the original volume-ratio inequality established by Ball and its Lp version established by Lutwak, Yang and Zhang.In Chapter 3, the minimization problem of Orlicz surface area is solved. An affine geometric quantity, called minimal Orlicz surface area, is introduced, which generalizes Petty’s minimal surface area and LYZ’s minimal Lp-surface areas. The bounds of mini-mal Orlicz surface area are estimated. Consequently, a reversed isoperimetric inequality is established for minimal Orlicz surface area. This affine inequality generalizes the reversed isoperimetric inequality established by Ball.In Chapter 4, the first Orlicz variations of quermassintegrals are studied. A class of geometric quantities, called Orlicz mixed quermassintegrals, are introduced, which gen-eralize the mixed quermassintegrals introduced by Aleksandrov, Fenchel and Jessen, and the Lp-mixed quermassintegrals introduced by Lutwak. The Cauchy-Kubota formula for Orlicz mixed quermassintegrals is proved, which associates the Orlicz Brunn-Minkowski theory with integral geometry. The Minkowski type isoperimetric inequality for Orlicz mixed quermassintegrals is established. Moreover, the Orlicz Brunn-Minkowski inequal-ity for quermassintegrals is also established.In Chapter 5, the first Orlicz variations of affine quermassintegrals are studied. A class of geometric quantities, called Orlicz mixed affine quermassintegrals, are in-troduced. In light of some integral geometric techniques on the Grassmann manifold, the affine invariance of Orlicz mixed affine quermassintegrals is proved in extenso. The Minkowski type isoperimetric inequality for Orlicz mixed affine quermassintegrals is established. Moreover, the Orlicz Brunn-Minkowski inequality for affine quermassinte-grals is also established.