Lower and Upper Bounds for Chord Power Integrals of Ellipsoids

First we discuss different representations of chord power integrals Ip(K) of any order p ≥ 0 for convex bodies K ⊂ Rd with inner points. Second we derive closed-term expressions of Ip(E(a)) for an ellipsoid E(a) with semi-axes a = (a1, . . . , ad) in terms of the support function of E(a) and prove upper and lower bounds expressed by the volume and the mean breadth of E(a), respectively. A further inequality conjectured in Davy (1984) is proved for ellipsoids. Some remarks on chord power integrals of superellipsoids and simplices round off the topic. Mathematics Subject Classification: Primary 52A40 60G05 ; Secondary 52A07 52A22

For any p ≥ 0 we define the pth-order chord power integral (CPI) of K by (with 0 0 := 0), where (x, u) := {x + α u : α ∈ R} stands for the line in direction u ∈ S d−1 through x ∈ R d and K|u ⊥ is the orthogonal projection of K on u ⊥ (= (d − 1)-dimensional subspace orthogonal to u).CPI's are of considerable interest in integral and stochastic geometry for a long time, see [11], [13], and have many applications in material sciences, physics and image analysis, see e.g.[4], [5], [14] and references therein.In textbooks of integral and convex geometry, see e.g.[11], [13], the r.h.s. of ( 1) is mostly written as integral w.r.t. the line measure µ (d) 1 (•) (defined on the space A(d, 1) of one-dimensional affine subspaces of R d ): where, for integers p = 2, . . ., d, the Blaschke-Petkantschin formula, see [11] (p. 363) provides the representations for k = 1, . . ., d with the motion-invariant k-flat measure µ (d) k (•) (defined on the space A(d, k) of k-dimensional affine subspaces of R d ) satisfying the normalization µ 1) for p = 0, 1 and (3) for k = d − 1 we get the following relations, see e.g.[12], . .determine a unique distribution function F µ,K (of the length L µ,K of the µrandom chord of K) which, however, does not characterize the shape of K completely, see [11].
Due to F. Piefke [9], see also [12], the r.h.s. of (1) can be expressed for any p > 1 by the distribution of the interpoint distance of two randomly chosen points inside K leading to for any real p > 1, i.e., the ratio I p (K)/V (K) 2 takes the form and the integral on r.h.s of the last line is known as (d + 1 − p)-energy of the probability measure U K (= uniform distribution on K).

CLT for a Class Poisson Cylinder Processes
To motivate our study of CPI's we state a central limit theorem (CLT) for the total volume of the union of isotropic Poisson k-cylinders included in an expanding convex domain K as ↑ ∞ , where K ⊂ R d is a convex body containing the origin o of R d as inner point.To be precise we need some further notation.For details and the proof of the below CLT the reader is referred to [7].
Let Π λ = {P i } i≥1 be a stationary Poisson point process on R d−k with positive intensity λ := E#{i ≥ 1 : where each point P i is associated with an independent copy (Ξ i , Θ i ) of some generic pair (Ξ 0 , Θ 0 ) which consists of a random compact set Ξ 0 ⊂ R d−k satisfying 0 < EH d−k (Ξ 0 ) 2 < ∞ and a random orthogonal matrix Θ 0 whose (uniform) distribution is induced by the normalized Haar measure on the Grassmannian of k-dimensional subspaces in R d .In addition, the components Ξ 0 and Θ 0 are independent and the sequence The countable familiy of random closed sets in R d is observed in an unboundedly increasing convex window K as ↑ ∞, see Fig. 1 and Fig. 2 for realizations of Ξ 2,1 and Ξ 3,1 with K = [0, 1] 2 and K = [0, 1] 3 , respectively.Now we are in a position to formulate the announced CLT for the volume fraction of Ξ d,k in K for k = 1, . . ., d − 1: As ↑ ∞, the random variable is asymptotically normally distributed with mean zero and variance where This CLT is of interest from several points of view.The Gaussian limit of (5) depends on the shape of the observation window, i.e. on K, expressed in terms of the CPI I k+1 (K).This shape-dependence of σ 2 (K) is caused by the intrinsic long-range correlations of the random set Ξ d,k .In contrast to this, in case of random sets satisfying certain weak dependence conditions, e.g. as in the degenerate case k = 0, the asymptotic variance σ 2 (K) depends only on the volume V (K).Note that Ξ d,0 can be identified with a stationary Boolean model with typical grain Ξ 0 , see [13] (p. 117).6) if another ovoid functional of K, e.g. the mean breadth b(K), is fixed.In the planar case optimal lower bounds of I 2 (K)/H 2 (K) 2 have been obtained for particular classes of convex sets in [6] when the perimeter H 1 (∂K) is given.In convex geometry, see [3] or [13], one is also interested to maximize I k+1 (K) when V (K) is fixed.Among all convex bodies the ball with radius (V (K)/κ d ) 1/d is the unique maximizer due to Carleman's inequality In the main part of the paper we study CPI's of d-dimensional ellipsoids E(a) with positive semi-axes a = (a 1 , . . ., a d ) defined by 3 A Formula for CPI's of Ellipsoids From ( 8) it is easily seen that the diagonal matrix of the convex body K, see [13] (p. 600), with twice continuously differentiable boundary ∂K can be expressed by the integral [1].In the latter formula the integrand is the sum of the d−1 j principal j-minors of the Hessian matrix Basic facts on support functions and their role in convex geometry can be found in [1] and [13].For example, the mean breadth b The support function of the ellipsoid E(a) defined in ( 8) is well-known and D d−1 (H E(a) (u)) can be expressed (after rather lengthy calculations) by some power of h E(a) (u) for u ∈ S d−1 , more precisely, This yields an integral expression of I 0 (E(a)) = κ d−1 S(E(a))/2.The remaining CPI's of E(a) are given in Theorem 3.1 For any real p ≥ 0, we have Proof.To begin with we rewrite (4) with indicator functions w.r.t. the ellipsoid K = E(a) so that, for any p > 1, By substituting x = A s and y = A t for s, t ∈ B d and applying the integral transformation formula twice the double integral on the r.h.s. of ( 11) takes the form Next, we introduce spherical coordinates z = r u for r = z ≥ 0 and u It remains to express the first integral in the last line by values of the Γfunction.Arguing from a purely geometric view point (including Cavalieri's principle), see [6] (p. 326), we find that Summarizing the foregoing steps confirms (10) for p > 1.The case p ∈ [0, 1] must be considered separately.From a formal point of view the term p − 1 occurring in the above formulas disappears by cancelling and in view of the relation Γ(p/2) p/2 = Γ(p/2 + 1) the r.h.s. of (10) makes sense for p = 0 (and even for p > −2).To be rigorous we consider the r.h.s.'s of ( 1) and ( 10 For reasons of symmetry we can reduce the integral over S 1 on the r.h.s. of (1) to four integrals over the quadrant [0, π/2] leading to After a short calculation, where h( φ) can be replaced by h(ϕ), we arrive at It should be noted that (12) was given (without proof) in [14] for p = 0, 1, . .., see also [6] for p = 2, and the chord length distribution function F µ,E(a,b) has been derived in [10].Equating (10) for d = 2 and (12) provides relations for complete elliptic integrals which are of interest in their own right.
Finally, we say a few words in an attempt to use (3) to calculate I k+1 (E(a)).
For doing this, we need formulas for the k-volume of the intersection k-ellipsoids E(a)∩L for L ∈ A(d, k).For k = d−1 we succeeded in obtaining such a formula by exploiting the fact that the (d−1)-ellipsoids E(p, u) Finally, after some further rearrangements we arrive at
Conjecture 1.For any convex body K ⊂ R d with inner points and k ≥ 0, with "=" being attained only for balls, where 2 κ k κ d−1 κ d+k κ k+1 κ d κ d+k−1 > 1 for k ≥ 1.To get an explicit lower bound of I k+1 (K) (as conjecture) we multiply on both sides of ( 14) over k = 1, . . ., n (after that we set n = k) and replace I 1 (K) by d κ d V (K)/2.In this way together with (7) we get the inclusion for k = 1, . . ., d, where for k = 0 both sides of ( 15 It should be noticed the fact that the inequality induced by the lower and upper bound in (15) coincides with the well-known isoperimetric inequality , where "=" holds iff K is a ball.
This means that the validity of ( 15) would present an (almost) optimal estimate of I k+1 (K) and slightly strengthen the isoperimetric inequality.
In what follows we prove the inclusion (15) for K = E(a) and derive in this case at least for k = d − 1 slightly larger lower bound in terms of b(E(a)).
Theorem 5.1 For the ellipsoid K = E(a) defined in (8) the Conjecture 1 and therefore the inclusion (15) are true.Furthermore, the inclusion The r.h.s. of (18) remains unchanged if In this way we may write the ratio of the both integrals over . Now, we insert this identity in the desired inequality (14) for K = E(a).Comparing the resulting relation ( 14) with (18) we deduce that ( 14) is equivalent with the moment inequality Since Y is strictly positive and bounded, it easily seen by the Cauchy-Schwarz inequality that for j = 0, 1, . . ., k − 1 and k = 1, 2, . .., implying immediately (19).Note that "=" holds iff P(Y = h −1 E(a) = const) = 1, i.e., a 1 = • • • = a d = const.Thus, the first part of Theorem 5.1 is proved.To prove the second part we start with 1 = ( h E(a) (u) Y (u) ) 1/2 for u ∈ S d−1 and apply the Cauchy-Schwarz and twice the Hölder inequality leading to for any real p ∈ [1, d].Next we write E U as integral over S d−1 w.r.t.uniform distribution U and make use of the mean breadth b(E(a)) = 2 E U h E(a) (see in Sect.3) with support function (9).This amounts to the inclusion Combining (10) for p = 1 and I 1 (K) = d κ d V (K)/2 reveals the remarkable relation We mention that (21) can also be verified directly by using the spherical coordinates j=i sin ϑ j for i = 3, . . ., d with infinitesimal surface element , where ϑ 1 ∈ [0, 2π] and ϑ i ∈ [0, π] for i = 2, . . ., d − 1.
Finally, we multiply the inclusion (20) by the constant C d,p which is chosen in view of (10) such that the middle term multiplied by C d,p just equals I p (E(a))/V (E(a)) 2 .This completes the proof of Theorem 5.1.2 Remark 2. From (17) we get Urysohn's inequality, see [1], for E(a), namely,  1) ).
This gives rise to formulate the following alternative to (15): Conjecture 2. For any convex body K ⊂ R d with inner points and k ≥ 0, with "=" being attained only for balls, see [6] for the special case k = d − 1.

CPI's of Superellipsoids and Simplices
We conclude this paper with a few remarks on CPI's of two important classes of convex bodies in R d -superellipsoids E α (a) and simplices S(P).To be precise, we start with a definition of these bodies.In generalizing (8) the convex body To conclude with, we quote a result on CPI's of S(P) proved in [15]  An astonishing relation proved in [8] says that H d−1 (S(P)|u ⊥ ) ρ DS(P) (u) = d V (S(P)) for any u ∈ S d−1 , where ρ DK (u) := max{λ > 0 : λ u ∈ K ⊕ (−K)} denotes the radial function of the difference body DK := K ⊕(−K) of K.It is not difficult to show that ρ DK (u) = max{H 1 (K ∩ (x, u)) : x ∈ u ⊥ } = length of the longest chord of K in direction u.In view of this geometric relation the integrand on the r.h.s. of (23) can be replaced by ρ DS(P) (u) p−1 , see also [2] for a different approach.

1
Chord Power Integrals -Definition and Basics Let K be a convex body in R d with interior points and S d−1 = ∂B d the boundary of the Euclidean unit ball B d = {x ∈ R d : x ≤ 1}.Further, let H k denote the k−dimensional Hausdorff measure on R d for k = 1, . . ., d.As usual let us denote by V (K) = H d (K) and S(K) = H d−1 (∂K) volume and surface content of K, respectively.Further, we recall that κ d := V (B d ) = π d/2 /Γ(d/2 + 1) and S(B d ) = d κ d with Γ(s) := ∞ 0 e x x s−1 dx for s > 0.
forms a stationary and isotropic process of Poisson k-cylinders in R d for k = 1, . . ., d − 1.The motion-invariant union set of the Poisson k-cylinders

Figure 1 :
Figure 1: Isotropic Poisson strips Figure 2: Poisson cylinders in a cube r) of the unit ball depends only on r ≥ 0 and disappears for r > 2 we find together with the infinitesimal transformation rule dz = r d−1 dr H d−1 (du) and A z = r A u = r h E(a) (u) that