Extremal Theories For Convex Polytopes And Their Applications

This Ph. D. dissertation sketches firstly the growing history, researching status quo, main represent figures, and works of mathematicians of our country in the researching branch; the following, it studies emphasisly the one open problems of the well-known Schneider's projection problem, next to research the Upper Bound Theorem and the Lower Bound Theorem called as the Dehn-Sommerville Relation, following to research the Minkowski inequality and the Brunn-Minkowski inequality for p-Quermassintegral differences of convex bodies and for mixed projection bodies, finally to research certain extension of Hilbert's Double-series inequality and gain the two theorems.The author has abtained the following results blazed new trails:(i) Some progress for Schneider's projection problem has been gained. To study the well-known Schneider's projection problem, in 2001, E. Lutwak, D. Yang and G. Zhang introduced a new affine invariant functional for convex polytopes in R~n. For origin-symmetric convex polytopes, they posed an open problem for the new functional. The auther give affirmative answers to the conjecture(the open problem) in origin-symmetric convex polytopes for H_n and give an application.(ii) Some progress for the classical combinatorial theory of convex polytopes has been grained. The main achievements in the modern theory of convex polytopes are the Upper Bound Theorem and the Lower Bound Theorem called as the Dehn-Sommerville Relations, these belong to the classical combinatorial theory of convex polytopes. The author established two exeremal theorems little know for symmetric polytopes, they may be seen as the Upper Bound Theorem and the Lower Bound Theorem of the metric theory of convex polytopes. In addition, some applications are given.(iii) The author studied Minkowski inequality and Brunn-Minkowski inequality for convex bodies and Mixed projection bodies, and established the Minkowski inequality and the Brunn-Minkowski inequality for p-Quermassintegral differences of convex bodies, which were extensions and strengthening of the classical Minkowski inequality and Brunn-Minkowski inequality. Further, gained the Minkowski inequality and the Brunn-Minkowski inequality for Quermassintegral differences of mixed projection bodies, which were general and strengthened forms of the Minkowski inequality and the Brunn-Minkowski inequality for mixed projection bodies.(iv) The author established two new inequalities similar to certain extension of Hilbert's Double-series inequality, which is avail to resolve the practical problem in engineering such as the study of earthquake.