A Result Of The Average Distance From The Points Within Convex Domain To The Boundary Points

Integral geometry is a subject that studies the properties of the graphics through all kinds of integral,and it belongs to differential geometry. Not only is it closely linked to the probability and statistics, but also biological, medicine, mineralogy, metal science. At the same time it has a widely application in physical, astronomical, architectural and acoustics .One of the earliest mathematician of China in the study of Integral geometry is WuDaRen. He extended the basic conclusion (including main formula) of Euclidean space in Integral geometry to 3 d elliptic space first. Then he also proved a series of inequality of chord-power integrals of convex body in E~2 and E~3. RenDeLin first gained the measure formula of certain line segment contained in convex body in 3 d Euclidean space. He also extended inequalities of the chord-power integrals in n d Euclidean space. At the same time he generalized Buffon needle problem.Convex geometric analysis is a subject that studies geometric structure and invariants of convex sets by using both geometric and analytic methods. In it's geometric properties, it not only belongs to a branch of the convex geometry, but also closely contact with differential geometry and integral geometry. The Brunn-Minkowski theory also called mixed volume theory is a hot issue in convex geometric analysis. The main results of Brunn-Minkowski theory is Brunn-Minkowski inequality, the solution of Brunn-Minkowski problem, The Aleksandrov-Fenchel inequality, Hadwiger,s characterization theorem. In addition, affine geometry of convex set is the branch of the integral of convex geometric analysis. In affine geometry of convex set, affine isoperimetric inequality and reverse isoperimetric inequality play a dominant role. Convex geometric analysis is a very useful discipline, it not only can be used in the differential geometry, integral geometry, algebraic geometry, but also plays a very important role to Monge-Ampere equation, number theory, Banach space theory, theory of probability research.This paper discussed the hot problem of convex geometric analysis- mixed volumes problem, and summarized some Minkowski inequality at first. Then, we mainly discussed the average distance between the interior point and the boundary point of convex domain and the average chord length in Integral geometry. Using the concept of the largest chord function and the support function, we gained the average distance formula between the interior point and the boundary point of parallelogram .Based on that formula we got the average distance formula between the interior point and the boundary point of rectangular, square and rhombus.