Chord-power Integrals And Chord Integrals

In this paper,we study the chord—power integrals and the double chord-power integrals of a convex body in R2,as well as the chord integrals of a,star body in Rn(n≥2). We use analysis method of the integral geometry to obtain some in-equalities for chord-power integrals and inequalities for double chord-power integrals In addition,we study the chord integrals of a star body and get some inequalities about them.The full text is divided into four parts:In the first part of this paper,we state the development process and research status of integral geometry and convex geometric analysis. We also introduce the main representatives and the works of scholars in our country in this respect.In the second part of this paper,we study the chord-power integrals and the double chord-power integrals of a convex body. We obtain some inequalities for chord-power integrals and inequalities for double chord-power integrals. The main result is:Theorem2.2If K is a Convex body in R2,Iet M=max{|p(Φ)||Φ∈[0,2π]), then In(K)≤2n+l,B2-n(K).Theorem2.5(The geNeralization of Holder inequality) Im-1,0/p-n(K)Im-1,0/p-m(K)≥Im-1,0/p-m(K), where m,n,p satisfy0≤m≤n≤p. Theorem2.7If If is a convex body in R2, m is an even number. Then we have this inequality for double chord-power integral where,ri is the radius of the maximum inscribed disk Bi:re is the radius of the minimum external disk Be.In the third part of this paper, we use the relation between inclusion mea-sure and the chord-power integrals, the integral formula for chord-power integral of elliptic domain is obtained. The main result is:Theorem3.2The integral formula for chord-power integral of elliptic domain K is:In the fourth part of this paper, we study the chord integrals of a star body. We get some limit relations of chord integrals. We also obtain some inequalities for chord integrals, including the inequalities between chord integrals and dual quermass-integrals, the dual Blaschke-Santalo inequality. The main result is:Theorem4.2If K∈So/n,i≠n,there is limit formula as followsTheorem4.3If K∈So/n,μ>0, j=0,1.2,…, n-1, Kμ=K+μB, thenTheorem4.9If K∈So/n,i≠n,then Pn+i(K)n≤ωn/n-iV(K°)i, with equality holds if and only if K is a ball centered on the origin. Theorem4.10(The dual Blaschke-Santalo inequality) If K€So/n,then V(K)V(K°)≥ωn/2, with equality holds if and only if K is a ball centered on the origin.