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Chord length distribution and the distance between two random points in a convex body in R

In this article for n-dimensional convex body D the relation between the chord length distribution function and the distribution function of the distance between two random points in D was found. Also the relation between their moments was found.


Introduction
The integral geometric concepts such as the distribution of the chord length, the distribution of the distance between two random points in a convex body D and many others carry some information about D. In this article the relation between the chord length distribution function and the distribution function of the distance between two random points uniformly distributed in a convex body was found. By R n we denote the n-dimensional Euclidean space (here we assume that n > 2, for case n = 2 see [1]). Let D ⊂ R n be a compact convex set (convex body) with boundary ∂D. By V n (D) we denote the ndimensional volume of D. By S n−1 we denote the unit sphere in R n centered at the origin O and let e O,ξ be the hyperplane containing O and normal to ξ ∈ S n−1 . By G n we denote the set of all lines in R n . We use the usual parametrization of a line γ = (ω, P): where ω ∈ S n−1 is the direction of γ and P is the intersection point of γ and e O,ω . By [D] we denote the set of lines intersecting D. In G n we consider the invariant measure (with respect to the group of Euclidean motions) µ (·). It is known that the element dγ of the measure, up to a constant, has the following form ( [2], [4], [6]) here dω and dP are elements of the Lebesgue measure on the sphere and the hyperplane, respectively. By χ (γ) we denote the length of a chord D ∩ γ. We consider γ as a random line intersecting D with nor- . The distribution function F χ (u) of χ (γ) is called the chord length distribution function of D. Thus we have Now let Q 1 = (x 1 , ..., x n ) , Q 2 = (y 1 , ..., y n ) be random independent points uniformly distributed in D and denote by d distace between Q 1 and Q 2 . By F d (t) we denote the distribution function of d. We have here dQ i (i = 1, 2) is the usual Lebesgue's measure in R n .

Main results
Now we present the main results. The following theorem describes the relation between F χ (t) and F d (t).
Theorem 2.1. Let D be a convex body. We have the following relation between F χ (t) and F d (t) Also in the article was found the following relation.

Theorem 2.2.
Let D be a convex body. The following relation between the k-th moment of the distance between two random points and the moments of the chord length distribution of D is valid

Preliminary results
In this section we need to prove the following lemmas.
where D ω is the orthogonal projection of D onto hyperplane e O,ω . In this article, we consider a convex body D with positive Gaussian curvature at every point of ∂D. For ξ ∈ S n−1 we denote by s (ξ) the point on ∂D the outer normal of which is ξ. It is known that (see [3], [5]) here ds (ξ) is the element of n − 1-dimensional Lebesgue's measure on ∂D. Substituting (3.3) into (3.2) and using the Fubini's theorem we obtain For any ξ ∈ S n−1 we have (see [3]) For a line γ intersecting a convex body D we have the following lemma.
For any ω ∈ S n−1 it is obvious that χ (ω, P) dP is the element of n-dimensional volume of D, hence the integrating by dP over D ω we get V n (D). [D] It is known that a pair of points Q 1 , Q 2 in R n can be represented by the line γ = (ω, P) passing through the points and pair of one dimensional coordinates (t 1 , t 2 ) (see [6]). Thus (Q 1 , Q 2 ) = (γ, t 1 , t 2 ) = (ω, P, t 1 , t 2 ).
(3.10) Lemma 3.3. The Jacobian of the transform (3.10) is Proof of lemma 3.3. For a fixed Q 1 we represent Q 2 by polar coordinates with respect to Q 1 . It is known that where r = |Q 1 − Q 2 | and ω is the direction of Q 2 − Q 1 . For a fixed ω the point Q 1 can be represented by P and t 1 . Thus and by multiplying (3.12) and (3.13) and taking into account that r = |t 1 − t 2 | we get (3.14)

Proof of theorem 2.1
By definition of the distribution function we have Using (3.11) from lemma 3.3 we get We represent the integral by the form After calculating J 1 we obtain that for χ (γ) t and J 1 (t) = 2 (χ (γ)) n+1 n (n + 1) (4.6) for χ (γ) < t.
Now we denote by J 2 , J 3 , J 4 the following integrals I {χ(γ) t} dγ, (4.8) I {χ(γ)<t} (χ (γ)) n+1 dγ. (4.10) Easy to notice (using (3.1) from lemma 3.1) that To calculate J 3 we change it the following way and calculate it instead. First we calculate the derivative of J 5 and then intergrate from 0 to t (J 5 (0) = 0). Using the first mean value theorem for definite integrals we have and after integrating that we get For J 3 we have Using the same technique we can calculate J 4 and get that After substituting (4.11), (4.16), (4.17) into (4.7) finally we obtain that Note that in R 2 formula (4.19) first was found in [1]. Now we are going to prove theorem 2.2.