The Containment Problem And Bonnesen-Type Inequalities In The Euclidean Space R～n

The isoperimetric inequality,that is:Let A and L be the area and the perimeter of any domain D in the Euclidean space R²,then satisfy the inequality L²-4Ï€A≥0,with the equality holds if and only if D is a disc.In 1920,Bonnesen obtains stronger isoperimetric inequalities,that is L²-4Ï€A≥π²(r_e-r_i)²,where r_i and r_e be,respectively,the radiuses of the biggest inscribed circle and the smallest circumscribed circle of domain D.In 2004,Jiazu,Zhou get a series of Bonnesen-type inequality in Euclidean space R².In 2006,Jiazu,Zhou and Fangwei,Chen obtain a series of Bonnesen-type inequality in a plane of constant curvature.Isoperimetric inequality is famous Alexandrov-Fenchel inequality in higher dimension space.In this thesis,we obtain the Bonnesen-type inequality that is the low bounds of the integral of square of mean curvature∫_(?)KH²gσabout convex body K.Many mathematician has been interested in obtain sufficient condition to guarantee that a given domain D₁ of surface area A₁,bounded by a simple piece-wise smooth boundary(?)D₁,of volume V₁ may be contained in another given domain D₀ of surface area A₀,bounded by a simple piecewise smooth boundary (?)D₀,of volume V₀.The type of condition sought is meaningful if it just depends on volumes V₁,V₀,surface areas A₁,A₀,curvature integrals of boundaries(?)D₁ and(?)D₀ of the two domains involved.In 1941,Hadwiger(see[11,12])was the first to use the method of integral geometry to obtain some sufficient conditions for one domain to contain another in the Euclidean space R² by estimating the kinematic measure of one domain moving into another under the rigid motions in R².In 1988,Gaoyong,Zhang(scc[11,18])obtains sufficient condition for one convex domain to contain another in the Euclidean space R³ by estimating the kinematic measure of one convex moving into another under the rigid motions in R³;in 1994,Jiazu,Zhou(see[21])obtains sufficient condition for one convex do-main to contain another in the Euclidean space R⁴ by estimating the kinematic measure of one domain moving into another under the rigid motions in R⁴;and in1995.Jiazu,Zhou(see[4,11,21,22])obtains sufficient conditions for one domain to contain another by estimating the kinematic measure of one domain moving into another under the rigid motions in R³ and in 3-dimension constant curva-ture space.In this thesis,we obtain a series of Bonnesen-type inequality in the Euclidean space R³,R⁴,and Rⁿ,by using the method and idea of Jiazu,Zhou and result of containment measure of Gaoyong,Zhang and Jiazu,Zhou.