Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows

In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.


Introduction
The problem of reconstruction of a convex body from the mean and Gauss curvatures of the boundary of the body goes back to Christoffel and Minkwoski [1].Let F be a function defined on 2-dimensional unit sphere 2  S .The following problems have been studied by E. B. Christoffel: what are necessary and sufficient conditions for F to be the mean curvature radius function for a convex body.The corresponding problem for Gauss curvature is considered by H. Minkovski [1].W. Blaschke [2] provides a formula for reconstruction of a convex body B from the mean curvatures of its boundary.The formula is written in terms of spherical harmonics.
A. D. Aleksandrov and A. V. Pogorelov generalize these problems for a class of symmetric functions , G R R of principal radii of curvatures (see [3]- [5]). .In n-dimensional case, a Christoffel-Minkovski problem is posed and solved by Firay [6] and Berg [7] (see also [8]): what are necessary and sufficient conditions for a function F, defined on 1 S n− to be function ( ) ( ) convex body, where 1 1 p n ≤ ≤ − and the sum is extended over all increasing sequences 1 , , p i i  of indices chosen from the set 1, , 1 i n = −  .R. Gardner and P. Milanfar [9] provide an algorithm for reconstruction of an origin-symmetric convex body K from the volumes of its projections.D. Ryabogin and A. Zvavich [10] reconstruct a convex body of revolution from the areas of its shadows by giving a precise formula for the support function.
In this paper, we consider a similar problem posed for the projection curvature radius function of convex bodies.We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.The solution of the system of differential equations is itself interesting.Let 3 ⊂ B R be a convex body with sufficiently smooth boundary and with positive Gaussian curvature at every point of the boundary ∂B .We need some notations.Note that one can lead the problem of reconstruction of a convex body by projection curvatures using representation of the support function in terms of mean curvature radius function (see [7]).The approach of the present article is useful for practical point of view, because one can calculate curvatures of projections from the shadows of a convex body.Let's note that it is impossible to calculate mean radius of curvature from the limited number of shadows of a convex body.Also let's note that this is a different approach for such problems, because in the present article we lead the problem to a differential equation of spatial type on the sphere and solve it using a new method (so called consistency method).
The most useful analytic description of compact convex sets is by the support function (see [11]).The support function of B is defined as ( ) we denote the restriction of H onto the circle S ω for 2 S ω ∈ , and call the restriction function of H. Below, we show (Theorem 1) that Problem 1. is equivalent to the problem of existence of a function H defined on 2  S such that ( ) for every 2 S ω ∈ .
Definition 1.If for a given F there exists H defined on 2 S that satisfies Equation (1), then H is called a solu- tion of Equation (1).

( )
H ω ϕ is a function defined on the space of an ordered pair orthogonal unit vectors, say , e e , (in integral geometry such a pair is a flag and the concept of a flag was first systematically employed by R.V. Ambartzumian in [12]).
There are two equivalent representations of an ordered pair orthogonal unit vectors 1 2 , e e , dual each other: is the spatial direction of the first vector 1 e , and ϕ is the planar direction in S ω coincides with the direction of 2 e , while 2 S Ω ∈ is the spatial direction of the second vector 2 e , and Φ is the planar direction in S Ω coincides with the direction of 1 e .The second representation we will write by capital letters.
(no dependence on the variable Φ ), then G is called a consistent flag solution.
There is an important principle: each consistent flag solution G of Equation (1) produces a solution of Equation (1) via the map and vice versa: the restriction functions of any solution of Equation ( 1) onto the great circles is a consistent flag solution.
Hence, the problem of finding a solution reduces to finding a consistent flag solution.
To solve the latter problem, the present paper applies the consistency method first used in [13]- [15] in an integral equations context.
Equation ( 1) has the following geometrical interpretation.It is known (see [11]) that 2 times continuously differentiable homogeneous function H defined on where ( ) is the restriction of H onto S ω .So in case 0 F > , it follows from (7), that if H is a solution of Equation (1) then its homogeneous extension is convex.
It is known from convexity theory that if a homogeneous function H is convex then there is a unique convex body 3 ⊂ B R with support function H and ( ) , F ω ϕ is the projection curvature radius function of B (see [11]).
The support function of each parallel shifts (translation) of that body B will again be a solution of Equation (1).By uniqueness, every two solutions of Equation ( 1) differ by a summand , a ⋅ defined on 2  S , where = , where R is the projection curvature radius function of B (see [16]).The purpose of the present paper is to find a necessary and sufficient condition that ensures a positive answer to both Problems 1,2 and suggest an algorithm of construction of the body B by finding a representation of the support function in terms of projection curvature radius function.This happens to be a solution of Equation (1).
Throughout the paper (in particular, in Theorem 2 that follows) we use usual spherical coordinates , ν τ for points 2  S based on a choice of a North Pole and a reference point 0 τ = on the equator.The point with coor- dinates , ν τ we will denote by ( ) , ν τ , the points ( ) 0,τ lie on the equator.On S ω we choose anticlockwise direction as positive.On the plane ω ⊥ containing S ω we consider the Cartesian x and y-axes where the direc- tion of the y-axis y  is taken to be the projection of the North Pole onto ω ⊥ .The direction of the x-axis x we take as the reference direction on S ω and call it the East direction.Now we describe the main result.
Theorem 2 Let B be a 3-smooth convex body with positive Gaussian curvature at every point of ∂B and R is the projection curvature radius function of B. Then for 2 S Ω ∈ chosen as the North pole is a solution of Equation ( 1) for F R = .On S ω we measure ϕ from the East direction.Remark, that the order of integration in the last integral of (8) cannot be changed.Obviously Theorem 2 suggests a practical algorithm of reconstruction of convex body from projection curvature radius function R by calculation of support function H. We This follows from Equation (1), see also [16].b) For every direction 2 S Ω ∈ chosen as the North pole where the function F * is the image of F (see ( 3)) and y is the direction of the y-axis on ( ) Let F be a positive 2 times differentiable function defined on  .Using (8), we construct a function F defined on 2 S : ( ) Note that the last integral converges if the condition (10) is satisfied.(9), (10) and the extension (to 3 R ) of the function F defined by (11) is convex.

The Consistency Condition
We fix 2 S ω ∈ and try to solve Equation (1) as a differential equation of second order on the circle S ω .We start with two results from [16].a) For any smooth convex domain D in the plane ( ) ( ) ( ) where ( ) h ϕ is the support function of D with respect to a point s D ∈ ∂ .In (12) we measure ϕ from the normal direction at s, ( ) R ψ is the curvature radius of D ∂ at the point with normal direction ψ .b) ( 12) is a solution of the following differential equation One can easy verify that (also it follows from ( 13) and ( 12)) is a flag solution of Equation ( 1).Theorem 4 Every flag solution of Equation (1) has the form where n C and n S are some real coefficients.Proof of Theorem 4. Every continuous flag solution of Equation ( 1) is a sum of 0 G g + , where 0 g is a flag solution of the corresponding homogeneous equation: for every 2 S ω ∈ .We look for the general flag solution of Equation ( 16) in the form of a Fourier series After substitution of ( 17) into ( 16) we obtain that ( ) 0 , g ω ϕ satisfies ( 16) if and only if ( ) ( ) ( ) Now we try to find functions C and S in (15) from the condition that g satisfies (4).We write ( ) where ( ) G ω ϕ was defined in (14).
Here and below ( ) Φ ′ ⋅ denotes the derivative corresponding to right screw rotation around Ω. Differentiation with use of expressions (see [14]) sin , tan sin , cos , cos after a natural grouping of the summands in (18), yields the Fourier series of ( ) ( ) . By uniqueness of the Fourier coefficients

Averaging
Let H be a solution of Equation ( 1), i.e. restriction of H onto the great circles is a consistent flag solution of Equation (1).By Theorem 1 there exists a convex body ∈  B with projection curvature radius function R F = , whose support function is H.
To calculate ( ) we take Ω for the North Pole of 2 S .Returning to the Formula (15) for every ( ) We integrate both sides of ( 22) with respect to uniform angular measure dτ over [ ) Now the problem is to calculate We are going to integrate both sides of (20) and (21) with respect to dτ over [ ) Integrating both sides of (20) and ( 21) and taking into account that i.e. a differential equation for the unknown coefficient ( ) We have to find ( ) given by (24).It follows from (27) that ( ) ( ) .cos cos Integrating both sides of (5.1) with respect to dν over [ ) Fixing τ and using (32) we write a Taylor formula at a neighborhood of the point π 2 Substituting ( 33) and (34) into (30) and taking into account the easily establish equalities ( ) ( ) Using expressions (19) and integrating by dϕ yields From (44), using (9) we obtain (8).Theorem 2 is proved.

Proof of Theorem 3
Necessity: if F is the projection curvature radius function of a convex body ∈  B , then it satisfies (9) (see [16]), the condition (10) (Theorem 5) and F defined by (11) is convex since it is the support function of B (Theorem 2).
Sufficiency: let F be a positive 2 times differentiable function defined on  satisfies the conditions (9), (10).We construct the function F on 2 S defined by (11).There exists a convex body B with support function F since its extension is a convex function.Also Theorem 2 implies that F is the projection curvature radius of B.
radii of curvature of the boundary of B at the point with outer normal direction

2 SR. 1 .Problem 2 .
the great circle with pole at 2 S ω ∈ , ( ) ω B -projection of B onto the plane containing the origin in 3 and orthogonal to ω , ( ) call projection curvature radius of B.Let F be a positive continuously differentiable function defined on the space of "flags" In this article, we consider: Problem What are necessary and sufficient conditions for F to be the projection curvature radius function Reconstruction of that convex body by giving a precise formula for the support function.

2 S
Ω Φ (dual each other).Let G be a function defined on  .For every 2 S ω ∈ , Equation (1) reduces to a differential equation on the circle S ω .Definition 2. If ( ) , G ω ⋅ is a solution of that equation for every ω ∈ , then G is called a flag solution of Equation (1).Definition 3. If a flag solution ( ) We denote: [ ] ,e Ω Φ -the plane containing the origin of 3 dual to ( ) , ω ϕ .Note that in the Problem 1. uniqueness (up to a translation) follows from the classical uniqueness result on Christoffel problem, since turn to Problem 1.Let R be the projection curvature radius function of a convex body B. Then F R

) Theorem 5 2 S
For every 3-smooth convex body ∈  B and any direction .For the case where there are two or more directions of maximum one can apply a similar argument.Now we take the point O * of the convex body B for the origin of3 Let F be a positive function defined on  .If Equation (1) has a solution H then there exists a convex body B with projection curvature radius function F, whose support function is H. Every solution of Equation (1) has the form 3 a ∈ R .Thus we have the following theorem.Theorem 1