Extremal Problems And Stability In Geometric Analysis

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 two open problems of the well-known Schneider's projection problem regarding projection bodies and Bourgain's problem for isotropic bodies in the local theory of Banach space; the third, it discusses the extremal properties for the polars of mixed projection bodies and cases of Pythegroean inequalyties on John basis; finally, it consideres the metric of homothetic deviation of m(m 2) geometric bodies (star bodies or convex bodies) and the deviation metric which is called the deviation regular metric of a simplice, and obtains stability theorems of some class geometric inequalities for simplices .The author has obtained the following results blazed new trails:(i) A real breakthrough 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 Rn. For origin-symmetric convex polytopes, they posed an open problem for the new functional, the author given affirmative answers to the conjecture(the open problem) in R2 and R3, The author established a recursion formula for the affine-invariant in Rn. Thereby, a real breakthrough for Schneider's projection problem has been gained.(ii) Bourgain's problem for isotropic bodies has been solved partly. Bourgain's problem, finding the least upper bound of isotropic constant LK of convex bodies K, is a well known open problem in the local theory of Banach space. The best estimate known is LK < cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body K Rn. Utilizing the method of spherical section function, the author has proven that if K is a convex body with volume one and r1B2n C K C r2B2n, (r1 1/2,r2 n/2), then LK < 1 and the conditions with equality have been found. Thereby, Bour-gain's problem for isotropic bodies has been solved partly.(iii) Pythagorean inequalities for convex bodies on John basis have been gained. In 1960, W. J. Firey established Pythagorean inequality for the mixed volumes of convex bodies on an orthogonal basis. The author established Pythagorean inequalities for convexbodies on John basis.(iv) A new stability version for the dual Aleksandrov-Fenehel inequality has been established. For the dual Aleksandrov-Fenehel inequality, Gardner and Vassllo established a stability version in 1999. Following their works, the author introduced a metric method for homothetic deviation of m, (m 2) geometric bodies (star bodies or conevx bodies), and established a new stablity version for the dual Aleksandrov-Fenehel inequality by the metric mthod and a refinement of Holder's inequality.(v) Stability theorems of Euler's and Weitzenbock's inequalities for simplices have been obtained. A new deviation metric which is called the deviation regular metric of a simplice is introduced. Utilzing the deviation metric, the author established stability theorems of Euler's and Weitzenbock's inequalities for simplices.