Research On Some Inequalities In Convex Geometry Analysis

This paper belongs to the Brunn-Minkowski theory，which devoted to studying somerelated extrme value and inequailites of convex bodies，star bodies and other geometricobjects. The main technologies used in this paper are basic notions，basic theories andintegral transforms in the Brunn-Minkowski theory and its dual theory.The main results can be stated as follows:(1) The infimum of the dual quermassintegral product of convex bodies and its polaris obtained，which is dual to the infimun of quermassintegral product of convex bodiesand its polar obtained by Lutwak.(2) A cyclic inequality and a monotonic inequality of Lp-mixed quermassintegralsare established. Meanwhile，we obtain an inequality for Lp-mixed quermassintegrals ofconvex bodies and its polar.(3) A new type of Brunn-Minkowski inequality for mixed affine surface area isestablished. As applications，we obtain two new Brunn-Minkowski inequalities for dualquermassintegrals of star bodies. Futhermore，we extend the equality condition ofBrunn-Minkowski inequality for the affine surface area of curvature images.(4) The notion of general Blaschke bodies is introduced. Some basic propertiesabout the general Blaschke bodies are presented. Further，the extremum values of thevolume and affne surface area for general Blaschke bodies are established respectively.As applications of this work，we give the negative answers to the Shephard problem forthe volume and affne surface area respectively，which extend the results obtained byPetty and Schneider.(5) Two inequalities are established for the mixed chordintegrals of star bodiesanalogous to the cycle inequality and Brunn-Minkowski inequality for the width-integralsof convex bodies. Furthermore， we present an inequality associating the mixedchord-integrals with dual quermassintegrals.(6) We establish the Lp-Brunn-Minkowski type inequalities for radial Blaschke Minkowski homomorphisms，which in special cases yield some of new results forintersection bodies. Meanwhile，we obtain two monotonic inequalities for radial BlaschkeMinkowski homomorphisms.