The Minkowski Quermassintegrale of the Outer Parallel Convex Body 

Author  ZhaoXueHua 
Tutor  ZhouJiaZu 
School  Guizhou Normal 
Course  Computational Mathematics 
Keywords  Trigonometric identities Convex sets Convex body Convex surface Curvature Surface area Volume Orthogonal projection Quermassintegrals Outer parallel convex body Parallel surfaces Mean curvature integral 
CLC  O186.5 
Type  Master's thesis 
Year  2007 
Downloads  30 
Quotes  0 
Homogeneous integral Minkowski proposed, is a very important concept in the theory and integral geometry of convex bodies and tools. Kubota, Cauchy, Steiner and many predecessors Quermassintegrals not given in a series of formulas and theorems. Quermassintegrals description convex body K and its orthogonal projection K '_{ nr } the convex body parallel to the outer convex body is one of the important concepts in integral geometry In this article, we give Several properties of the orthogonal projection of the convex body parallel to the outer convex body. Several properties are given in [O] (nr) the dimension space L _{ nr (nr)dimensional convex The bodies K ' nr Quermassintegrals W' i (K ' nr ) and K' nr outer parallel convex body (K ' nr ) ρ surface area F ((K' nr ) ρ ), volume V ((K ' nr ) ρ ), border (?) ((K' nr ) ρ ) the average curvature integral M i ((?) ((K ' nr ) ρ )) and ndimensional convex body in ndimensional space K homogeneous integral relationship between the first part of this paper several trigonometric identities, these identities prove and promote differential geometry the classic Euler formula when they will be used in differential geometry and integral geometry but as trigonometric identities, these formulas suitable for the more general case. Theorem 1 Let the three angles θ } the θ _{ i the θ j to meet: θ = ± (θ i θ j ), or θ = θ i the θ j , or θ θ i < / sub> θ j = 2π, there are formulas in Note 1 of this theorem is applied to the proof of Euler's formula, is an elementary proof of Euler's formula process. Theorem 2 when θ = ± ( θ i θ j ), when θ = θ i θ j or θ θ i 0 , j = 2π, Note 2 special applications as these trigonometric identities, Chen Zhao and Zhou gives a simple proof of Euler's formula and its analogy (see reference [2]) the second part of this article is on the outer parallel convex body Minkowski Quermassintegrals Kubota, Cauchy, Steiner and others are given a series of formulas and theorems, Wuhan University Li Zefang on this part of the done a lot of work, her job is the limit K ' nr parallel to the outer convex body made plane L nr [O] Here is her conclusion: Conclusion Let K for the ndimensional Euclidean space the E n convex body, K ρ the K distance ρ outer parallel convex body, (K ρ ) ' nr K ρ the outer parallel convex body made (nr) the dimension plane L nr [O] ( (K ρ ) ' nr is K ρ (nr)dimensional plane L nr [O] the orthogonal projection). W r 1 j (K) is a convex body K in ndimensional Euclidean space E n Quermassintegrals, F ((K ρ ) ' nr ) said (?) ((K ρ )' nr ) of the surface area (ie, (nr 1)dimensional volume), Conclusion Let K be the ndimensional Euclidean space E n in the convex body, K ρ is the the K distance ρ outer parallel convex body, (K ρ ) ' nr K ρ (nr)dimensional plane L nr [O] made on the outer parallel convex body (ie (K ρ ) ' nr K ρ L nr (nr)dimensional plane [O] on orthogonal projection). W (ri) (K) the homogeneous integral convex body K in ndimensional Euclidean space E n , V ((K < sub> ρ ) ' nr ) (K ρ )' nr (nr)dimensional volume Conclusion 3 Let K be the the n the Vea space E n convex body, K ρ K the distance to the the ρ outer parallel convex body, (K ρ ) ' nr is K ρ parallel convex body's outer (nr) the dimension plane L nr [O] (( K ρ ) ' nr K ρ positive (nr)dimensional plane L nr [O] orthogonal projection). the W rs 1 j (K) is a convex body K in the Veio space E n n Quermassintegrals M S ((?) (K ρ ) ' nr ) said (?) ((K ρ )' nr ) sth mean curvature integral, the main work of the second part of this article is divided into two areas: on the one hand is the Kubota formula to promote it from the (n1)dimensional space L extended to n1 [O] (nr)dimensional space L nr [O] in; work the other to the Lize Fang made K ' nr convex body parallel to the outer limit L nr [O] plane this restriction removed, but parallel convex body in ndimensional space E n for outside (nr)dimensional space to promote the ndimensional space. following definition and Lemma parallel convex body in the second part of this article we in the Minkowski Quermassintegrals of used: 1 (convex sets, convex body, convex surface ) Let K be an ndimensional Euclidean formula the space E N in a subset of, if when A, B ∈ K, connecting two points of the line segment AB is also belong to K, K is called E convex sets. compact convex set has a nonempty interior is called a convex body. boundary of the convex body K (?) Note 3 in the future, we limit the discussion to a bounded convex body. 4 O n ndimensional area of ??the unit sphere, can be expressed as follows: Lemma 1 n dimensional Euclidean space E n in a fixedpoint nondirectional the total measure of the rdimensional plane (ie: the Grassmann manifold G r nr volume) which O i is the area of ??the idimensional unit sphere defined (the Minkowski a fixed point in the qualitative Points) Let K n  dimensional the spaces E n convex body O the E n . L nr [O] represents any one of the points O (nr) dimensional plane viewed K at each point perpendicular to the L NR [O] rdimensional plane, the rdimensional plane and L nr [O] intersection constitute a convex body K ' nr K' nr called K to L nr [O] orthogonal projection, K ' nr the (nr)dimensional volume denoted by V (K' nr ). because over the fixed point (nr)dimensional plane L nr [O] and form a Grassmann Manifold G NR R we introduce the following integral this equation r = 1, ..., n1 is given I r < / sub> (K) defined. addition, supplementary provisions: I 0 (K) = V (K) (K ndimensional volume) (7) Grassmann manifold G nr r is the volume of the formula (5), we obtain the integral average of the projection K 'of the volume NR V (K' nr ): E n convex body K Minkowski Quermassintegrals of W r (K) is defined as follows: 1. when r = 1, ..., n1, or 2 when r = 0 when 3 when r = n, define 3 (outer parallel convex body, parallel surfaces) set K is an ndimensional Euclidean the space E N in a convex body, to each point in K for the core constant ρ radius as a closed sphere, the sphere and set the distance called K ρ outer parallel convex body denoted by K the ρ K ρ boundary (?) K to the ρ called boundary (?) K distance ρ parallel surfaces Definition 4 (mean curvature integral) Let Σ ndimensional Euclidean the spaces E n a C 2 class hypersurface k 1 k 2 , ..., k n1 Σ (n1) of a principal curvature (function), respectively, the the r th average curvature of Σ points (denoted as M r (Σ)) is defined as: wherein d σ Σ the area element, {the k i 1 the k i 2 , ..., k the i l } main curvature rth order elementary symmetric functions. addition, supplementary provisions: of M 0 (Σ) = F (ie Σ the product of the area). (14) k 1 k 2 ... k n1 called the curved surface of the GaussKronecker curvature and surfaces The area of ??the spherical aberration yuan du n1 contact: d σ is the area of ??the surface Σ yuan R i = 1 / (k i ) (i = 1, ..., n1) called Σ main radius of curvature. average curvature Points can be the main curvature radius defined as follows: where {R i < sub> 1 , R i 2 , ..., R i nr1 i >} for the R i the 1 2 , ..., R the i n convex 1 first (n1)rorder elementary symmetric function Lemma 2 (Kubota formula) Let K be the ndimensional Euclidean space E n body, L n1 [O] represents any one of the point O (n1)dimensional plane, K ' n1 for the K in L projection on the n1 [O] of. W r (K) for the ndimensional Euclidean space E n of the convex body K Quermassintegrals W ' r1 (K' n1 ) in the (n1)dimensional space L n1 [O] convex body K ' n1 the Quermassintegrals. (1/2) U n1 represents half of the (n1)dimensional unit sphere , that G 1, n1 ; Q n2 (n2)dimensional area of ??the unit sphere Lemma 3 (Cauchy formula) Let K for the ndimensional a convex body in the Euclidean space E n , (?) K K convex surface, the W 1 (K) of K Quermassintegrals, F is (?) K in surface area (i.e. (n1)dimensional volume), then F = nW 1 (K) (16) Lemma 4 (Steiner formula) Let K is an ndimensional Euclidean space E < sub> n convex body, K ρ the K distance ρ outer parallel convex body V (K the ρ ) K the volume ρ the W j (K) the K Quermassintegrals, Lemma 5 Let K for the ndimensional Euclidean space E n in a convex body, K ρ for the distance K is ρ parallel to the outer convex body, W ij (K) the the K Quermassintegrals, W i (K ρ ) K ρ Quermassintegrals Lemma 6 Let ndimensional Euclidean space E n body K boundary (?) K is a C 2 class hypersurface the the K ρ for K the distance ρ outer parallel convex body (?) K is (?) a distance of K parallel surfaces is ρ the W r 1 (K) of K Quermassintegrals, M r ((?) K) is (? ) the mean curvature Points of K the M i ((?) K ρ ) (?) the K ρ mean curvature integral, The use of the above lemma, we can get the following theorem: Theorem 3 Let K for the ndimensional Euclidean space the E n convex body, K ' nr K (nr orthogonal projection on) the dimension plane L nr [O / sub>. W ir (K), said K in ndimensional Euclidean space E n the Quermassintegrals W ' i (K' nr ) (nr)dimensional space L nr [O] convex body K ' nr Quermassintegrals O i is the area of ??the idimensional unit sphere, G r, nr said Grassmann manifold. Note 5 when r = 1, Theorem 3 is the Kubota the formula. Theorem 4 Let K for the ndimensional Euclidean space the E n convex body, K ' nr K to orthogonal projection on the (nr)dimensional plane L nr [O] , (K ' nr ) ρ for K' nr, distance is ρ in ndimensional Euclidean space E n parallel to the outer convex body. W j 1 r (K) represents the convex body K in ndimensional homogeneous integral in Euclidean space E n , F ((K ' nr ) ρ ) (?) ((K' NR ) ρ ) in the surface area (i.e. (n1)dimensional volume), wherein the O i is the area of ??the idimensional unit sphere, G r, nr said Grassmann manifold. Theorem 5 Let K ndimensional Euclidean space E n convex body K ' nr , K to (nr) the dimension plane L nr orthogonal projection on [O] , (K ' nr ) ρ K' nr ndimensional Euclidean space E n distance ρ parallel to the outer convex body. W jr (K) convex body K in n Dimensional style homogeneous integral in the space E n , V ((K ' nr ) ρ ) (K' nr ) ρ ndimensional volume, O i is the area of ??the idimensional unit sphere, G r, nr said Grassmann manifold Theorem 6 Let K is ndimensional Euclidean space the E n convex body, K ' nr the K dimensional plane (nr) L nr [O] orthogonal projection , (K ' nr ) ρ K' nr ndimensional Euclidean space E n distance ρ outer parallel convex body (?) ((K ' nr ) ρ ) (?) (K' nr ) distance ρ is parallel to the surface, and (?) ((K ' nr ρ ) ) C 2 class hypersurface. W ij 1 r (K) convex body K in ndimensional Euclidean space E n Quermassintegrals M i < / sub> ((?) ((K ' nr ) ρ )) (?) ((k' nr ) mean curvature integral ρ ), O i is the area of ??the idimensional unit sphere, G r, nr said Grassmann manifold. }