Extreme Problems In Convex And Discrete Geometry

This thesis deals with the extreme problems in the Brunn-Minkowski theory and the L_p—Brunn-Minkowski theory. Some extreme problems in discrete geometry are also investigated.Except the first chapter, the thesis can be divided into the following three parts:(I) We study some extreme problems in the Brunn-Minkowski theory, such as the Loomis-Whitney inequality, Schneider projection problem and some extreme properties of simplex. We establish the Loomis-Whitney inequality for mixed body and give the affirmative answer of the Schneider projection problem for regular polygon. This part consists of the second chapter, the third and the fourth chapter.(II) In this part (the fifth chapter), we deal with the extreme problems in the L_p—Brunn-Minkowski theory. Some inequalities which are dual to the corresponding results given by E.Lutwak, D.Yang and G.Zhang are established.(III) The last part consists of the sixth and the seventh chapter. Here, we study some sort of Heilbronn's problems and establish some results which strengthen the Soifer' corresponding results. Finally, a interesting inequality is established which is the generalization of the well-known arithmetic-geometric mean inequality.