Mixed Brightness-Integrals Of Convex Bodies

The mixed width-integrals of convex bodies are defined by E. Lutwak in 1977. In this paper, the mixed brightness-integrals of convex bodies are defined. An in-equality is established for the mixed brightness-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (in-volving the mixed brightness-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened version of this general in-equality is obtained by introducing indexed mixed brightness-integrals.Lutwak, Yang and Zhang introduced the notion of Lp-centroid body, Ma Tongyi posed the definition of Lp-mixed harmonic Blaschke add. In the paper, combining the Lp-mixed harmonic Blaschke add and Lp-mixed centroid body, Brunn-Minkowski inequality for the quermassintegrals and dual quermassintegrals of Lp-mixed Centroid Body is established.In Chapter 1, we give a brief introduction to significance and current situation in the study of convex geometry.In Chapter 2, we present some elementary results of convex geometry.In Chapter 3, in the first, half the brightness is defined.And mixed brightness-integrals of convex bodies. Besides that, the properties of the mixed brightness-integral are listed. An isoperimetric inequality (involving the mixed brightness-integrals) is presented which generalizes an inequality recently obtained by Chake-rian and Heil. Strengthened version of this general inequality is obtained by intro-ducing indexed mixed brightness-integrals.In Chapter 4, combining the Lp-mixed harmonic Blaschke add and Lp-mixed centroid body, Brunn-Minkowski inequality for the quermassintegrals and dual quer-massintegrals of Lp-mixed Centroid Body is established.In Chapter 5, we make a summary of the whole paper and put forward some problems for further study.