MEASURE OF PLANES INTERSECTING A CONVEX BODY

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1 Sutra: International Journal of Mathematical Science Education c Technomathematics Research Foundation Vol. 3 No. 1, pp. 1-7, 1 MEASURE OF PLANES INTERSECTING A CONVEX BODY RAFIK ARAMYAN Russian-Armenian (Slavonic) University, Institute of Mathematics Armenian Academy of Sciences, Bagramian 4b, Yerevan, ARMENIA rafikaramyan@yahoo.com In the present paper a representation for the measure of planes intersecting a convex body, using stochastic approximation of the body was found. The representation was found in terms of the normal curvatures of the surface of the body and a flag density of the measure. Keywords: integral geometry; stochastic approximation; flag densities. 1. Introduction Let E be the space of planes in R 3. We consider locally finite signed measures μ in the space E, which possess densities with respect to the standard Euclidean motion invariant measure, i. e. (see. [?]) μ(de) =h(e) de. (1) Recall that an element de of the standard measure is written as de = dp dξ, where (p, ξ) is the usual parametrization of a plane e: p is the distance of e from the origin O; ξ S is the direction normal to e, dξ is an element of solid angle of the unit sphere S. Where appropriate we write h(e) =h(p, ξ). The concept of a flag in R 3 which naturally emerges in Combinatorial integral geometry will be of basic importance below. A detailed account of this concept is in [?]. We repeat the definition. A flag is a triad f =(P, g, e), where P is a point in R 3 called the location of f, g is a line containing the point P,ande is a plane containing g. Therearetwo equivalent representations of a flag: f = f(p, Ω, Φ) or f = f(p, ω, ϕ), where Ω is the spatial direction of g in R 3, Φ is the rotation of e around g, ω is the normal of e, andϕ is the planar direction of g in e. The range of Ω and ω is 1

2 R. Aramyan, the standard elliptic space which can be obtained from the unit sphere by identification of the antipodal points ([?]), φ and ϕ belong to E 1. We introduce the following function in the space of flags (a flag function) ρ(f) =ρ(p, ω, ϕ) = cos (ϕ ψ) h [P ] (ξ) dξ. () Here [P ] is the bundle of planes containing the point P R 3, h [P ] (ξ) is the restriction of h onto [P ], ψ is the direction of the projection of ξ into the plane of the flag f. The notation h [P ] (ξ) is reasonable since ξ completely determines a plane from [P ]. Clearly, the integral () does not depend on the choice of the reference point on the plane of the flag f. The function ρ defined on the space of flags F we call flag density. The concept of a flag density was introduced and systematically employed by R. V. Ambartzumian (see [?],[?]). Note, that in [?] (seealso[?] and[?]) () was considered as an integral equation and by integral geometry methods was recovered h from a given ρ. Let B be a convex body with sufficiently smooth boundary and with positive Gaussian curvature at every point of B. By[B] wedenotethesetofplanesintersecting B. Lets(ω) beapointon Bwhose outer normal is ω. Byk 1 (ω),k (ω) we denote the principal normal curvatures of B at s(ω) andletk(ω, ϕ) be the normal curvature in the direction ϕ at the point s(ω) of B, ϕ is measured from the first principal direction. The main result of the paper is the following. Theorem 1.1. Let μ beasignedmeasureone, possessing a density h(e). For any sufficiently smooth convex body B we have the following representation: π μ([b]) = (π ) 1 k1 (ω)k (ω) ρ(s(ω),ω,ϕ) S k dϕ dω, (3) (ω, ϕ) where ρ is the flag density of μ defined by (). If h(e) 1 (the case of Euclidean motion invariant measure μ inv )from(3)we obtain the Minkowski formula (see [?]) μ inv ([B]) = 1 ( 1 k 1 (ω) + 1 ) dω. (4) k (ω) S. Preliminary Representations In [?] R. V. Ambartzumian has indicated the existence of the so-called flagrepresentation for width functions of convex bodies in R 3 using some standard flag-representation for width functions of polyhedra. Let H(ξ) be the width function in direction ξ of a convex body B. Then(see[?]) H(ξ) = sin α(ξ,ω, Φ) m(dω,dφ), (5) S 1 S

3 Measure of Planes Intersecting a Convex Body 3 where S i is the unit sphere in R i+1, i =1,, Ω, ξ S, m is a measure in S 1 S, α is the angle between Ω S and the trace e ξ e(ω, Φ), e ξ is a plane normal to ξ, ande(ω, Φ) is the plane of the so-called free flag f = f(ω, Φ) (one can consider that the location of a free flag is the origin of R 3 ). The representation (5) fails to be unique (there are many m for given H). Note, that if μ is a translation invariant measure in E with the form dμ = dp δ ξ, where δ ξ is a delta measure concentrated on the direction ξ, thenwehaveh(ξ) = μ([b]). Let K R 3 be a convex polyhedron and e [K] E. We consider the intersection e K which is a bounded convex polygon whose vertices correspond to the edges of K actually hit by e. The fact, that the sum of outer angles of e K equals π we write in the form α i (e) I [Li](e) =πi [K]. (6) i Here L i is an edge of K, α i (e) is the outer angle of e K correspond to vertex e L i, and summation is by all edges of K. In [?], by integration of (6) with respect to dμ = dp m(dξ) (a translation invariant measure) some standard flag-representation for the width function of a polyhedron K was found. In [?], using approximation by polyhedrons, a new representation for the width functions of convex bodies was obtained. In [?], using stochastic approximation (Voronoi s approximation) of smooth convex bodies by polyhedrons, for translation invariant measures representation (3) was obtained. Now we integrate (6) with respect to μ(de) = h(e)de, whereh is a continuous function defined on E. Wehave πμ([k]) = α i (e) h(e) de = α i (e) h(p, ξ) dp dξ i [L i] i [L i] = i [L i] α i (e)h(x, ξ) cos( ξ,ω i ) dxdξ = i L i α i (e)h(x, ξ) cos( ξ,ω i ) dξdx. Here ξ,ω i is the angle between ξ and Ω i,whereω i is the direction of the edge L i. Also, here we use the following well known fact from integral geometry de = dp dξ = cos( ξ,ω i ) dx dξ, (8) where x is the intersection point e L i and dx is one dimensional Lebesque in L i. Using standard formulae of spherical trigonometry we get (see [?]) α i (e) cos( ξ,ω i ) = sin α(ξ,ω i, Φ) dφ, (9) A i where A i is the exterior dihedral angle of the edge L i (see also (5)). After substitution (9) into (7) we obtain πμ([k]) = sin α(ξ,ω i, Φ) h(x, ξ)dξ dφ dx. (1) i L i A i (7)

4 4 R. Aramyan 3. Stochastic Approximation Let B be a sufficiently smooth (three times continuously differentiable) convex body in R 3. We assume that the Gaussian curvature of B is everywhere positive. Hence the Gauss map of B onto the unit sphere S is a homeomorphism. We throw n independent points P 1,...,P n onto S with the same distribution P. Let dp =f(ω)dω, wheref(ω) > is continuous, dω is an area element on S.On B by P1,...,P n we denote the images of the points P 1,...,P n by the inverse to the Gauss map. Denote by K n (P1,...,P n ) the convex hull of the points P 1,...,P n. According to (1), μ([k n (P1,...,P n )]) can be represented in the form n πμ([k n ]) = I D (i, j) sin α(ξ,ω ij, Φ) h(x, ξ)dξ dφ dx. (11) i<j i,j=1 L ij A ij Here Ω ij is the direction of Pi P j, L ij is the edge Pi P j, A ij is the exterior dihedral angle of the edge Pi P j, D is the set of all pairs (i, j) corresponding to the edge. We average both sides of (11) with respect to the sequences (P1,...,P n ). Since f(ω) >, in the limit (n )intheleft-handsideweobtainμ([b]). By symmetry we have ( ) [ n I D (1, ) n (S ) (S ) [ n ] ] sin α(ξ,ω 1, Φ) h(x, ξ)dξ dφ dx dp 3 dp n dp 1 dp. (1) L 1 A 1 Taking P 1 as the pole, P can be described by spherical coordinates (ν, ϕ) with respect to P 1.Also,P can be described by coordinates (l, ϕ), where l = P 1 P.We have dp = f(ω) dω = f(ν, ϕ) sinνdνdϕ= f(l, ϕ) ldldϕ. (13) Let e(ω lϕ, Φ) be the plane passing through P1,P and rotated around Ω lϕ = P 1 P by angle Φ. For e(ω lϕ, ) we take the plane that is perpendicular to the plane passing through ω and Ω lϕ.byl we denote the segment P1 P and let l = P1 P. In this paper we consider the case of the uniform distribution, i.e. f(ω) = ( C(ω)) 1,where is the total area of the surface of B, C(ω) is the Gaussian curvature at the point on B with normal ω. The plane e(ω lϕ, Φ) divides B into two parts and by S(Φ,l) we denote the area of the smaller part B 1 (Φ,l). Applying Fubini s theorem in the inner integral of (1), we obtain ( ) [ π [ ( n 1 S(Φ,l) ) n ( ) ] n S(Φ,l) + n (S ) π S o [ ] ] sin ldldϕdω α(ξ,ω lϕ, Φ) h(x, ξ)dξ dx dφ L So. (14) C(ω)C(l, ϕ) The sum in the square brackets of (14) is the probability that segment P1 P is an edge and e(ω lϕ, Φ) belongs to the exterior dihedral angle of the edge.

5 Measure of Planes Intersecting a Convex Body 5 Since S(Φ,l) lim n 1,wehave ( ) n (S ) [ π π ( ) n [ S(Φ,l) L ldldϕdω SoC(ω)C(l, lim ϕ) A n ] ] sin α(ξ,ω lϕ, Φ) h(x, ξ)dξ dx dφ ( )( ) n n 1 =, (15) where A is a constant. In a similar manner one can prove that the domain of variation of Φ and l can be taken arbitrarily small. Thus n ( ) n (S ) π l Φ [ L sin α(ξ,ω lϕ, Φ) h(x, ξ)dξ dx Φ ] ldφdl ( 1 S(Φ,l) ) n S o dϕ dω, (16) C(ω)C(l, ϕ) where l and Φ are arbitrarily small fixed numbers. From the regularity of the surface B we obtain the Taylor expansion S(Φ,l)=lS l (, ) + ΦS Φ (, ) + l S Φ ll (, ) + lφs lφ (, ) + S ΦΦ (, ) + R(Φ,l), (17) where R(Φ,l)=o(l +Φ ). Here all functions continuously depend on l and Φ, as well as on ω and ϕ. Below, we will see, that S l (, ) = S Φ (, ) =. Using the mean value theorem we find sin α(ξ,ω lϕ, Φ) h(x, ξ)dξ dx = l sin α(ξ,ω lϕ, Φ) h(x,ξ)dξ, (18) L where x is a point from the segment L and l = L. After substitution (18) in (16) and a change of variables u = l n, v =Φ n we get ( ) n n (S ) π [ sin v α(ξ,ω n u ϕ, ) h(x,ξ)dξ n l n Φ n Φ n (1 S( v n, u ] ] u(u b(ω, ϕ))du dv n C( v n,ϕ) ) n n ) dϕ dω S oc(ω), (19) where l = l b(ω, ϕ)+o(l). One can interchange the limit and the integration operations. Substitution (17) in (19) and making use of sin α(ξ,ω u n ϕ, v ) h(x,ξ)dξ sin α(ξ, ω, ϕ 1 ) h [P (ω)](ξ)dξ () n almost everywhere when n,whereα(ξ, ω, ϕ 1 ) is the angle between the direction ϕ 1 in the plane e ω and the intersection of e ξ with the plane e ω,[p (ω)] is the

6 6 R. Aramyan bundle of planes containing the point P (ω) B with normal ω, weobtain π [ ) πμ([b]) = exp ( u (S ) S ll(, ) uvs lφ(, ) v S ΦΦ(, ) u du dv ] [ ] b(ω, ϕ) sin α(ξ, ω, ϕ 1 ) h [P (ω)](ξ)dξ dϕ dω, (1) C(ω) where b(ω, ϕ) = lim l l l. 4. A Representation Obtained by Stochastic Approximation It follows from (1) that the final representation for μ([b]) depends on values of S ll (, ), S lφ (, ), S ΦΦ (, ) which are functions of ω and ϕ (see (1)). It was proved in [?], that: S l (, ), S Φ (, ), S ll (, ), S lφ (, ), S ΦΦ (, ) depend only on derivatives of at most order of two of the surface B at the point P whose outer normal is ω. Hence the corresponding calculation we can do for the osculating paraboloid of B at the point P (ω) whose outer normal is ω. In[?] the following expressions for the derivatives in terms of the normal curvatures of B at the point P (ω) were found: S l(, ) =, S Φ(, ) =, S ll(, ) = π k 1 k r (ϕ)(k 3 cos ϕ + k1 3 sin ϕ) A 4, S lφ (, ) = π k 1 k sin ϕ(k k 1 ) A 3, S ΦΦ (, ) = π k 1 k r(ϕ) A, () where k i, i =1, are the main normal curvatures, r(ϕ) =k1 1 cos ϕ + k 1 sin ϕ is the radius of the normal curvature in the direction ϕ at the point P (ω) of B and A = k cos ϕ + k1 sin ϕ. Also, in [?] was found that (see (1)) cos b(ω, ϕ) = ϕ k1 + sin ϕ k Substituting () and (3) into (1) we get π [ μ([b]) = (π ) 1 and tan ϕ 1 =tanϕ k 1 k. (3) ] sin k1 k α(ξ, ω, ϕ) h [P (ω)](ξ)dξ (S ) E k dϕ dω, (4) (ω, ϕ) where k(ω, ϕ) is the normal curvature in the direction ϕ at the point P (ω) of B. Taking into account that sin α(ξ, ω, ϕ) =cos (ϕ ψ), (5) where ψ is the direction of the projection of ξ into the plane with normal ω, weget (3). Theorem 1 is proved. I would like to express my gratitude to Prof. R. V. Ambartzumian for helpful discussions.

7 Measure of Planes Intersecting a Convex Body 7 References Ambartzumian, R. V. (1987). Combinatorial integral geometry, metric and zonoids. Acta Appl. Math., 9: 3-7. Ambartzumian, R. V. (199). Factorization Calculus and Geometric Probability), Cambridge University Press, Cambridge. Aramyan, R. G. (1986). About stochastic approximation of convex bodies. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), : Aramyan, R. G. (199). Recovering rose of directions from flag densities in R 3. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 7: Aramyan, R. G. (9). Solution of one integral equation on the sphere by methods of integral geometry. Doklady Mathematics, 79: Panina, G. Yu. (1986). Convex bodies and translation invariant measures. Zap. Nauchn. Semin. LOMI, 157: Ambartzumian, R. V., Ohanyan, V. K. (1994). Finite additive functionals in the space of planes. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 9: Santalo, Luis A. (1976). Integral Geometry and Geometric Probability, Addison-Wesley Publishing Company, Canada.