Several Problems's Studies In Integral Geometry

In this paper, we mainly investigate three problems, one is that the double chord-power integrals of a convex body in R^d. Secondly, by using characters of the flattened convex body's mean curvature integrals, we discuss about the mean curvature integrals of the outer parallel body of a projected convex set in E^d. Finally, we extend a formula of S. S. Chen.In the second chapter, we study the double chord-power integrals of a convex body in R^d. The concept of double chord-power integrals is a new concept on the base of chord-power integrals. The chord-power integrals is a special case of the double chord-power integrals. And double chord-power integrals get more geometric information. In this paper we obtain the following geometric inequalities:Theorem 2.5. Let K be a convex body in R^d, m, n are non-negative integer, thenTheorem 2.6. Let K be a convex body in R^d, m, n,p are non-negative integer and 0≤m≤n≤p, thenTheorem 2.7. The double chord-power integrals have the inequalitiesSpecial casewhen m > n,Theorem 2.8. The double chord-power integrals have the inequality Theorem 2.9. The double chord-power integrals have the inequalities:when n is integer.especially, when n is odd.when n is even.In the third chapter we discuss the mean curvature integrals of a projected convex set of the outer parallel body of in E^d. This is a interesting problem, Santal(?) ,Professor Zhou, Jiang Deshou, Li Zefang and so on investigate the problem, especially, Professor Zhou and Jiang Deshou study the mean curvature integrals of the outer parallel body of a projected convex set in E^d. Author study the mean curvatureintegrals of a projected convex set of the outer parallel body of in E^d. This are two different problems, the results gotten is different, we obtain the following theorem:Theorem 3.4. Let K be a convex body in E^d with C²-smooth boundary (?)K,K^r be projection on the r-plane L_r (?) E^d, and (K^r)_ρbe the outer parallel body of K^T in the distanceρin E_d. M_i^r((?)(K^r))(i = 0, l,…,r-1) be the mean curvature integrals of (?)(K^R) as a convex surface of K^R and let M_I^d((?)(K^r)_ρ, )(i = 0,1,..., d-1) be the mean curvature integrals of (?)(K^r)_p as a flattened convex body of E^d and (?)(K^r)_p∈C². Then we have1) If i < d - r - 1, then where V_r(K^r) denotes the r-dimensional volume of K^r.2) If i = d - r - 1, then3) If i> d-r-1, thenOtherwise, Letâ–³be the angle between two intersected linear subspaces through a fixed point O, the integral of the angleâ–³over the intersected subspace play an important role in integral geometry, this integral is basic problem in integral geometry: find explicit formulas of the integrals of geometric quantities over the kinematic density in terms of known integral invariants. In the fourth chapter, We extend an integral formula of S. S. Chen (the problem is belong to the above integra)and obtain an integral of n-power of the angle of two intersected linear subspaces. Following, we introduce the theorem:Theorem 4.1. Let L_q[0] be a fixed q-plane through a fixed point O and let L_p[0] be a moving p-plane through O.Assume that p + q > d. Letâ–³be the angle between the two linear subspaces, express dL_d-p[0]^2d-p-q be the density of dL_d-p[0] as a subspace of the fixed dL_2d-p-q[0], then we havewhere N is integer, O_i is the surface area of the i-dimension unit sphere.