Some Geometric Inequalities For Simplex And Their Applications

This thesis researches some important problems of geometric inequalities for simplexin convex geometry and distance geometry by using analytic method and theory of geo-metric inequality. Some geometric inequalities are established,and their applications aregiven.The thesis contains three chapters.Chapter 1 introduces development and significance of geometric inequality, espe-cially its development in china in last 20 years. And then, we give some basic conceptsconcerning simplex,and this thesis's main aspects at the end of chapter 1.Chapter 2 researches conjecture and improvement of n- dimension Euler inequality,Some geometric inequalities for the circumradius and iuradius of simplex and distance ofits interior point to side faces are estabished, which all are new improvements of Eulerinequality as application.Chapter 3 researches some geometric inequalities for volumes of simplex. A classof geometric inequalities for volumes involving two simplices and their interior points areestablished. In addition,we get some inequalities for volumes of a simplex and its pedalsimplex. As their applications, we can get a class of inequalities on the volumes of sim-plex and its pedal simplex, some known inequalities on the volumes of simplex and itspedal simplex are improved. In part 3,we obtain a result on the volume of its inscribedsimplex, as special case,we get n-dimensional Menelaus Theorem.In addition,we establishan inequality for the volumes of the tangent points simplex and the escenters simplex.