Research On Several Probabilistic And Geometric Inequalities

The central limit problem is one of the most interesting problems in the field of asymptotic geometry and has received a lot of attention from mathematicians as a bridge between probability theory and convex geometric analysis.On the one hand,it is closely related to probability theory,and the central limit theorem well known to people is the important content and one of the cornerstone of probability theory,and its theoretical results are relatively perfect.The study of the central limit theorem has long influenced the development of probability theory by forming the analytical approach to probability.Since its inception in the 18th century,the central limit theorem has had a significant impact on practical life,particularly in explaining and predicting phenomena that occur in real life(e.g.human height,wave height,shooting errors,and so on).On the other hand,Anttila,Ball and Perissinaki[1]studied the central limit problem for centrally symmetric convex bodies and proved that the thin shell-estimate implies an affirmative answer to the central limit problem.Their work made the central limit problem widely known among people working in convex geometry.An important concept in convex geometry is the centroid body,and the study of various properties of it has been a popular area of research in convex geometry,leading to many elegant isoperimetric inequalities for the centroid body.In addition,the centroid body is another key tool for dealing with the central limit problem.The first main focus of this paper is the central limit theorem,which investigates a generalized class of radial log-concave measures and proves that they are also solutions of the central limit problem.The main tool we use is the log-concave function,and by applying some of its properties flexibly and rationally,we obtain thin-shell estimates for this class of measures.This type of measure contains a q-th Gaussian moment measure,and its optimal thin-shell estimate is obtained by direct computation and concrete discussion,leading to the optimal Berry-Esseen bound,i.e.the speed of convergence.The second main focus of this paper is geometric functional inequalities of the Lp centroid body with respect to weights,studying weighted measures for convex bodies,extending the Lp Blaschke-Santaló inequality established by Lutwak and Zhang[2],obtaining a weighted polar Lp Busemann-Petty centroid inequality and characterizing the condition of equality(if and only if the convex body under consideration is a ball).The joint use of the weighted polar Lp Busemann-Petty centroid inequality and the Ball’s body yields a functional version of weighted polar Lp Busemann-Petty centroid inequality.