The Calculation On Chord Length Distribution Function Of Convex Domain

Integral Geometry is a subject which oringinated from the famous problem of Buffon needle.Another called of Integral Geometry is Geometric Probability, belonged to the category of differential geometry essentially, with the natures that investigating of a graphic through a variety of integral. The development has always been linked with geometric probability. The idea of Integral Geometry is to use the methods of probability to discuss convex body and geometry. The application and development have relationship with algebra,partial differential equation,geometric analysis,convex geometry,geometric inequality and so on.A lot of elderships made a great contribution to geometry,such as Santalo,Yan Zhida,Wu Daren,Ren Delin.Another important subject of convex body theory is the chord length distribution function of convex domain.It has much application background such as Pattern Recognition,Statistical analysis of materials.But now ,there is not provide a unified approach to obtain the chord length distribution function for convex domain in existing literature.In this paper we take triangle for example to discuss the approach to calculate the exact analytical formula of chord length distribution function in using generalization support function technique and limited chord function. This method can also be used in other areas.Convex Geometry is a branch of geometry,the study object is convex set or convex. In the 19th century, Hermann Brunn and Hermann Minkowski did a lot of pioneering work in the early development of the convex geometry. Brunn-Minkowski theory of convex geometry is a classic content .The core part is the Brunn-Minkowski theory and the theory of mixed volumes. Brunn-Minkowski theory has a profound link with many important branches of mathematics.In 1980s, E. Lutwak introduced the concept of dual mixed volume,further enriched the theory of convex bodies.And thus this solved many imporant issues which lacked of progress in a long term. At present this part of the convex geometry is still the most active research. Since the 1990s, China's mathematician Zhang Gao-Yong, has made a very excellent results in geometry inequality and ultimate solution of the Busemann-Petty problem.The geometric inequality which related with affine surface area is also important on the one hand.This paper discuss the proof in some of the inequalities under the existing finding of Zhang Gaoyong.