The Minkowski Quermassintegrale Of The Outer Parallel Convex Body

The concept of quermassintegrale is introduced by Minkowski.It is of basic importance in the theory of convex bodies and integral geometry.Kubato,Cauchy, Steiner and others give a serial of formulas and theorems about the quermassinte-grale.Quermassintegrale describe the relationships between the convex figure K and its orthogonal projection K'_n-r.In this Paper,we give several properties about the outer parallel convex body of the convex-figure's orthogonal projection.The properties respectively give the relations among quermassintegrale W_i+r(K)of the convex figure K,quermassintegrale W'_i(K'_n-r)of the orthogonal projection K'_n-rof K and the(n-1)-dimensional surface area F((K'_n-r)_ρ)and the n-dimensional volume V((K'_n-r)_ρ)of the outer parallel body(K'_n-r)_ρof the orthogonal projection K'_n-ras a convex figure in(n-r)-dimensional space L_n-r[O]and integral of mean curvature M_i((?)((K'_n-r)_ρ)).In the first part of the thesis,we give several triangle identical equations.These triangle identical equations describe the relationships amongθ_k(k=i,j)that are the angles between the curvature vector kβof the intersection curveΓof two surfaces∑_k(k=i,j),the two surfaces∑_k(k=i,j)and the angleθbetween∑_i and∑_j.As the triangle identical equations,these formulas are fit for the general conditions.Let the three anglesθ,,θ_i,θ_j satisfy the following equations:θ=±(θ_i-θ_j),orθ=θ_i+θ_j,orθ+θ_i+θ_j=2Ï€,thenWhenθ=±(θ_i-θ_j),we haveWhenθ=θ_i+θ_j,orθ+θ_i+θ_j=2Ï€,we getIn the second part of the thesis,we give several properties about the Minkowski quermassintegrale of the outer parallel body of the convex-figure's orthogonal pro-jection.Let K be a convex figure in E_n and let O be a fixed point.Let L_n-r[O]be all the(n-r)-planes through O and let K'_n-rbe the orthogonal projection of K into the L_n-r[O].If W_i+r(K)are the quermassintegrale of a convex figure K in E_n and W'_i(K'_n-r)are the quermassintegrale of K'_n-ras a convex figure in(n-r)-dimensional space L_n-r[O],then Let K be a convex figure in E_n and let O be a fixed point.Let L_n-r[O]be all the(n-r)-planes through O and let K'_n-rbe its orthogonal projection of K into the L_n-r[O].Denoted by(K'_n-r)_ρthe outer parallel body of K'_n-rin the distance p in E_n.If W_j+1+r(K)are the quermassintegrale of a convex figure K in E_n and F((K'_n-r)_ρ)denotes the(n-1)-dimensional surface area of(?)((K'_n-r)_ρ),thenLet K be a convex figure in E_n and let O be a fixed point.Let L_n-r[O]be all the (n-r)-planes through O and let K'_n-rbe its orthogonal projection of K into the L_n-r[O].Denoted by(K'_n-r)_ρthe outer parallel body of the orthogonal projection K'_n-rin the distanceρin E_n.If W_j+r(K)are the quermassintegrale of a convex figure K in E_n and V((K'_n-r)_ρ)denotes the n-dimensional volume of(K'_n-r)_ρ,thenLet K be a convex figure in E_n and let O be a fixed point.Let L_n-r[O]be all the (n-r)-planes through O and let K'_n-rbe its orthogonal projection of K into the L_n-r[O]and let(?)((K'_n-r)_ρ)be a hypersurface of class C² in E_n.Denoted by(K'_n-r)_ρthe outer parallel body of the orthogonal projection K'_n-rin the distance p in E_n. If W_i+j+1+r(K)are the quermassintegrale of K in E_n and M_i((?)((K'_n-r)_ρ))denotes the integral of mean curvatures of(?)((K'_n-r)_ρ),then...