The Average Distance Of Two Points Of A Kinematic Star Body On A Radial Direction

Integral geometry is a discipline investigating graphic properties by the means ofintegration, which essentially belongs to the category of the differential geometry. It derivesfrom geometric probability and its development also keeps contact with geometric probability.The kinematic formulas are central themes of integral geometry. There have been variousgeneralizations, variants and analogues of these formulas, partly motivated by their applications,which are from different points to depict the certain inherent attributes of objects in geometricspace.Professor Zhang Gaoyong established the kinematic formulas for dual quermassintegralsof star bodies by using dual mixed volumes and also gave a research on such a geometricprobability problem: Given two convex bodies in the space which have nonempty intersection,one is random, another is fixed. In general, the union of the two convex bodies is not convex.How close is it to be convex? Based on the solution of this problem and following the ideas andmethods to solve the problem, this paper takes use of the fundamental kinematic formulasinvolving quermassintegrals to the case of dual quermassintegrals and dual mixed volumes toget the following two conclusions:(1) According to definition of the average distance, the article gives the formule of theaverage distances between any two internal points of the intersection of two intersectingnon-empty convex bodies.(2) By introducing the concept of the linear radial combination for two convex bodies,with changing the formule of integrand, this paper studies the problem of the average distancesbetween any two internal points of their linear radial combination of two intersectingnon-empty convex bodies, obtaining the computional formulas of average distance.The article applies the kinematic formulas to geometric probability problems searching forthe average distance between between any two internal points of geometic objects, whichgeneralizes the relevant results of the existing dual kinematic formulas.