_{1}

^{*}

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.

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 [

A. D. Aleksandrov and A. V. Pogorelov generalize these problems for a class of symmetric functions

Let

R. Gardner and P. Milanfar [

D. Ryabogin and A. Zvavich [

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

Let F be a positive continuously differentiable function defined on the space of “flags”

Problem 1. What are necessary and sufficient conditions for F to be the projection curvature radius function

Problem 2. Reconstruction of that convex body by giving a precise formula for the support function.

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 [

The most useful analytic description of compact convex sets is by the support function (see [

Here

Given a function H defined on

Below, we show (Theorem 1) that Problem 1. is equivalent to the problem of existence of a function H defined on

for every

Definition 1. If for a given F there exists H defined on

In Equation (1),

There are two equivalent representations of an ordered pair orthogonal unit vectors

where

Given a flag function

where

Let G be a function defined on

Definition 2. If

Definition 3. If a flag solution

(no dependence on the variable

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 [

We denote:

where

Note that in the Problem 1. uniqueness (up to a translation) follows from the classical uniqueness result on Christoffel problem, since

Equation (1) has the following geometrical interpretation.

It is known (see [

where

So in case

It is known from convexity theory that if a homogeneous function H is convex then there is a unique convex body

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

Theorem 1 Let F be a positive function defined on

The converse statement is also true. The support function H of a 2-smooth convex body B satisfies Equation (1) for

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

Theorem 2 Let B be a 3-smooth convex body with positive Gaussian curvature at every point of

is a solution of Equation (1) for

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 turn to Problem 1. Let R be the projection curvature radius function of a convex body B. Then

a) For every

This follows from Equation (1), see also [

b) For every direction

where the function F^{*} is the image of F (see (3)) and

Let F be a positive 2 times differentiable function defined on

Note that the last integral converges if the condition (10) is satisfied.

Theorem 3 A positive 2 times differentiable function F defined on

We fix

a) For any smooth convex domain D in the plane

where

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

Proof of Theorem 4. Every continuous flag solution of Equation (1) is a sum of

for every

After substitution of (17) into (16) we obtain that

Now we try to find functions C and S in (15) from the condition that g satisfies (4). We write

where

Here and below

with use of expressions (see [

after a natural grouping of the summands in (18), yields the Fourier series of

the Fourier coefficients

where

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

To calculate

We integrate both sides of (22) with respect to uniform angular measure

Now the problem is to calculate

We are going to integrate both sides of (20) and (21) with respect to

where

Integrating both sides of (20) and (21) and taking into account that

for

i.e. a differential equation for the unknown coefficient

We have to find

Integrating both sides of (5.1) with respect to

Now, we are going to calculate

It follows from (15) that

Let

From (31), we get

Fixing

Similarly, for

Substituting (33) and (34) into (30) and taking into account the easily establish equalities

and

we obtain

Theorem 5 For every 3-smooth convex body

where

Proof of Theorem 5. Using spherical geometry, one can prove that (see also (1))

where H is the support function of B. Integrating (38), we get

Let

Theorem 6 Given a 2-smooth convex body

Proof of Theorem 6. For a given B and a point

Clearly,

It is easy to see that

Let

If ^{**} be the point for which

more directions of maximum one can apply a similar argument.

Now we take the point O^{*} of the convex body B for the origin of

By Theorem 6 and Theorem 5, we have the boundary condition (see (36))

Substituting (29) into (23) we get

Using expressions (19) and integrating by

where

and

Integrating by parts (42) we get

Using (34), Theorem 5 and taking into account that

we get

From (44), using (9) we obtain (8). Theorem 2 is proved.

Necessity: if F is the projection curvature radius function of a convex body

Sufficiency: let F be a positive 2 times differentiable function defined on

This work was partially supported by State Committee Science MES RA, in frame of the research project SCS 13-1A244.