Some Notes About Hadwiger's Containment Problem Of Convex Bodies In R~n And Bonnesen-type Inequalities

Reviewing the history of the integral geometry, convex geometry has always been studied as an important area. Convex bodies have lots of elegant nature, the researches make us discover and understand the connection of the geometric invariants of the domains involved. Probability and analysis are useful tools to study integral geometry. The sufficient conditions that one domain can contain another domain were studied, in 1942, Hadwiger worked out the case of plane and called Hadwiger's containment problem for convex bodies. But the analogue of Hadwiger's containment problem for higher dimensions were very complex, the conclusion isn't very perfect and continues develope. For example, in《The Willmore functional and the containment problem in R~4》, Jiazu Zhou obtained a new analogue of Hadwiger's condition for convex bodies in R~4, in《Geometric Inequalities and Inclusion Measures of Convex Bodies》, recently Gaoyong Zhang acquired a new analogue of Hadwiger's condition for convex bodies in R~n. The isoperimetric problem is also an important area of integral geometry, and it has very close relation with containment problem. In this thesis, we discuss another analogue of Hadwiger's condition for convex bodies in R~n and Bonnesen-type inequalities firstly. Then we proof some Bonnesen-type inequalities for surfaces of constant curvature. We obtain the area deficit and volume deficit by the isoperimetric deficit at last.We get the following theorems:Corollary 2.6. Suppose K and L be convex bodies in R~n, let A(K) be the area of K, V(K) the volume of K, V(L) the volume of L, M(L) the mean width of L. If V(K)≥V(L), then the following condition is sufficient to guarantee that L(?)K, up to an isometry,In the plane, this corollary is well-known Hadwiger's condition for convex sets.Theorem 2.10. If r_K and R_K are the inradius and outradius of a convex body K in R~n respectively, let A(K) be the area of K, V(K) the volume of K. ThenTheorem 3.1. Suppose K∈k(S_k~2), let P_K be the perimeter of K, A_K the area of K, r_K the inradius of K, R_K the outradius of K. Then the following inequality holds:Theorem 3.7. Suppose K∈k{H_λ~2), let P_K be the perimeter of K, A_K the area of K, r_K the inradius of K, R_K the outradius of K. Then the following inequality holds:with equality holds if and only if K is a hyperbolic disc.Theorem 4.1. Suppose A and V are the area and volume of a simply connected region D in R~n respectively. And let D~* be its convex hull with area A~* and volume V~*.Let△(D)=A~n-n~nω_nV~n be the isoperimetric deficit of D, whereω_n is the volume of the n-dimensional unit ball. Then we havewith the equality holds if D is a standard sphere.Theorem 4.2. Suppose A and V are the area and volume of a simply connected region in R~n respectively. And let D~* be its convex hull with area A~* and volume V~*. Let△(D)=A~n-nω_nV~n be the isoperimetric deficit of D, whereω_n is the volume of the n-dimensional unit ball. If A≥A~*, we havewith C is a constant and satisfy C<(?), the equality holds if D is a standard sphere.