On strict inclusions in hierarchies of convex bodies

Let $\mathcal I_k$ be the class of convex $k$-intersection bodies in $\mathbb{R}^n$ (in the sense of Koldobsky) and $\mathcal I_k^m$ be the class of convex origin-symmetric bodies all of whose $m$-dimensional central sections are $k$-intersection bodies. We show that 1) $\mathcal I_k^m\not\subset \mathcal I_k^{m+1}$, $k+3\le m<n$, and 2) $\mathcal I_l \not\subset \mathcal I_k$, $1\le k<l<n-3$.


Introduction
Let K and L be origin-symmetric star bodies in R n .Following Lutwak [L] we say that K is the intersection body of L if the radius of K in every direction is equal to the volume of the central hyperplane section of L perpendicular to this direction, i.e. for every ξ ∈ S n−1 , The closure in the radial metric of the class of intersection bodies of star bodies gives the class of intersection bodies.
A generalization of the concept of an intersection body was introduced by Koldobsky in [K3].Let 1 ≤ k < n and let K and L be origin-symmetric star bodies in R n .We say that K is a k-intersection body of L if for every (n − k)-dimensional subspace H ⊂ R n Vol k (K ∩ H ⊥ ) = Vol n−k (L ∩ H).The closure in the radial metric gives the class of k-intersection bodies, which will be denoted by I k .Note that I 1 is the class of intersection bodies.
Koldobsky [K2] introduced the concept of embedding of a normed spaces in L p , p < 0, and in [K3] he proved that k-intersection bodies are the unit balls of spaces that embed in L −k .
A well-known property of L p -spaces, proved in [BDK], is that, for any 0 < p < q ≤ 2, the space L q embeds isometrically in L p , so L p -spaces become larger when p decreases from 2. Koldobsky [K2] extended this result to negative p: Every n-dimensional subspace of L q , 0 < q ≤ 2, embeds in L p for every −n < p < 0.
However, it is an open problem, whether a normed space X = (R n , • ) being embedded in L −p for some 0 < p < n − 3 implies that X embeds in L −q for all p < q < n.In particular, is it true that every k-intersection body is also an m-intersection body for 1 < k < m < n − 3? Note that in some cases the above statement is known to be true.Since the product of positive definite distributions is also positive definite, one immediately obtains that if X embeds in L −p , 0 < p < n, and p divides q, p < q < n, then X also embeds in L −q ; see [M1].
2000 Mathematics Subject Classification.52A20, 52A21, 46B04.The author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.Part of the work was done when the author was visiting Université de Marne-la-Vallée.
Related to these are the questions of showing that these classes of bodies are different for different values of p and q.
It was shown by Koldobsky [K1], that there is an n-dimensional (n ≥ 3) Banach subspace of L 1/2 that does not embed in L 1 , (also a subspace of L 1/4 that does not embed in L 1/2 ).Later Borwein and his colleagues at the Center for Computational Mathematics at Simon Fraser University showed (by computer methods) that there is a Banach space that embeds in L a/64 but not in L (a+1)/64 for a = 1, 2,..., 63.Another construction was given by Kalton and Koldobsky in [KK] and allowed to extend these results to all 0 < p < q ≤ 1. Schlieper [S] used the construction from [K1] to show that there is a normed space that embeds in L −4 but does not embed in L −2 , and also a normed space that embeds in L −1/3 but not in L −1/6 .
In this paper we extend Schlieper's result to arbitrary integers, namely, we construct examples of origin-symmetric convex bodies which are k-intersection bodies, but not l-intersection bodies for 1 ≤ l < k < n − 3. We should remark that all origin-symmetric convex bodies are k-intersection bodies for k = n − 1, n − 2 and n − 3, see [K4, p. 78].
Another result that we present here is motivated by papers of Weil [W], Neyman [N] and Yaskina [Y].Weil constructed a convex body in R n (n ≥ 3) that is not a zonoid but all its projections onto hyperplanes are zonoids.Neyman showed that there are n-dimensional normed spaces that do not embed in L p , but all their (n − 1)-dimensional subspaces embed in L p for p > 0. Yaskina constructed a body in R n (n ≥ 5), which is not an intersection body, but all of its central hyperplane sections are intersection bodies.(Note, that all central sections of an intersection body are necessarily intersection bodies, see Fallert, Goodey and Weil [FGW]).
Here we generalize Yaskina's construction to prove the following.Let I m k be the class of convex bodies all of whose m-dimensional central sections are k-intersection bodies.There exists an origin-symmetric convex body One should compare this result with the fact (see e.g.[M1]), that all central sections of a k-intersection body are also k-intersection bodies provided that the dimension of the sections is greater than k.Therefore, I m+1 k ⊂ I m k .Finally let us remark that another generalization of intersection bodies was introduced by Zhang [Z].These are called generalized k-intersection bodies by Koldobsky and k-Busemann-Petty bodies by E. Milman.See [K3], [K4,Section 4.5], [M1], [M2] for many interesting results explaining the relation between these different generalizations and their connection to the lower dimensional Busemann-Petty problem.

Section:
Let us start with the following criterion for k-intersection bodies.
Theorem 2.1.(Koldobsky [K3]) Let K be an origin-symmetric star body in R n , The main result of this section is the following Theorem.
Theorem 2.2.Let k + 3 ≤ m < n.There exists an origin-symmetric convex body K that belongs to I m k , but not to I m+1 k .
Proof.For a small ǫ > 0 define a body K by , where |x| 2 is the Euclidean norm and E is the ellipsoid given by K is positive for a small ε, and so the body K is well defined.
The proof of the theorem follows from the following three lemmas.
Lemma 2.3.The body K is convex for small enough ε.
Proof.This is a standard perturbation argument, cf.[K4, p.96].By construction, the body K is obtained by perturbing the Euclidean ball.Since the latter has strictly positive curvature, it is enough to control the first and second derivatives of the function ǫ m−k x −k E .One can see that their order is O(ǫ m−k−2 ), which is small for small enough ǫ.Therefore K also has positive curvature.

Recall that the Fourier transform of |x|
where In order to compute the Fourier transform for the norms of ellipsoids, note that if T is an invertible linear transformation on R n , then Lemma 2.4.For every m-dimensional subspace H of R n , the body K ∩ H is a k-intersection body.
Proof.We have Since E is an ellipsoid with semiaxes ǫ and 1, E∩H is also an ellipsoid with semiaxes a 1 , ..., a m such that ǫ ≤ a i ≤ 1, ∀i = 1, ..., m.There is a coordinate system in H such that Taking the Fourier transform of y −k K∩H in the plane H we get Let a j be the smallest semiaxis.Then for some λ ≥ 1 we have a j = λǫ.Therefore, On the other hand if ξ ∈ S m−1 ⊂ H, then Therefore, ≥ 0 for all ξ ∈ S n−1 ∩ H and all H. Therefore all m-dimensional sections of K are k-intersection bodies.
Lemma 2.5.There exists an (m + 1)-dimensional section of K which is not a k-intersection body. .
The Fourier transform in the variables x 1 , ..., x m+1 equals Therefore K ∩ H is not a k-intersection body.
3. Section: We will need a few auxiliary lemmas.
Then the Fourier transform of the distribution f (θ)r −p , 0 < p < n, is a homogeneous degree −n + p continuous on R n \ {0} function, whose values on the unit sphere can be computed as follows.
(iii) is essentially from [Y,Lemma 2.4].For completeness we include a proof.For q close to 1 we use part (i) with k = 1 to get When q approaches 1, both the numerator and denominator in the right hand side tend to zero.Indeed, let us show that the limit of the numerator is zero: Recall the relation between the spherical Laplacian ∆ S and Euclidean Laplacian ∆ (see e.g.[G, p. 7]).If f is a homogeneous function of degree m, then on the sphere Due to the fact that ∆ S is a selfadjoint operator, [G, p. 7], we have In order to compute the limit of (1) as q → 0, apply l'Hopital's rule: Computing the Laplacian in the latter integral and using Euler's formula for derivatives of homogeneous functions, we get Using the relation between the Fourier transform and differentiation, and applying the latter formula to the function ∆ k−1 (f (θ)r −n+2k ) which is homogeneous of degree −n + 2, we have We will need the following spherical version of Parseval's formula, for the proof see [K4,Section 3.4].
Lemma 3.2.Let K and L be origin-symmetric infinitely smooth star bodies in R n and 0 < p < n.Then (2) In what follows C will always be a non-zero constant, not necessarily the same in different lines.We also use the notation a(ǫ) ∼ b(ǫ), meaning that lim is odd, then for every small α > 0 there exists a constant C α , such that for all ξ ∈ S n−1 , 1+α) .(iii) Moreover, in both cases, Applying ∆ under the integral, each time we get a factor of 1/ǫ 2 .This gives ǫ −2k = ǫ −n+p+q+1 .Finally use that x −1 E ≤ 1 for x ∈ S n−1 .To prove (iii) for this case, we use (3) again.Keeping only the terms of the highest order of ǫ, we get Consider the first integral in (4).As before, applying the Laplacian k times under the integral, we get a factor of ǫ −2k = ǫ −n+p+q .Therefore we need to estimate terms of the following form By Hölder's inequality, for a small α > 0, .
We use similar ideas to estimate the second integral in (4).Applying Parseval's formula two times and using the relation between the Fourier transform and differentiation, we get E ≤ 1 on the sphere.Combining the estimates for the both terms in (4), one can see that 1+α) .Now we will show that almost the same degree of dependence is achieved when ξ = e n .Indeed, using formula (4) with ξ = e n and dropping the second integral (which is small compared to the first integral), we get Formula (7) applied to the last integral gives = 2(−1) k After the change of the variable x n = ǫ • z, the latter equals where E * is defined by ( 6), and Note that after writing ln(ǫz) = ln ǫ + ln z, we will have two integrals, the first being equal to 2(−1) k ǫ ln ǫ Under the integral we have the Laplacian of a homogeneous function of degree −n + 2 which equals the spherical Laplacian.Since the spherical Laplacian is a self-adjoint operator, the latter integral is equal to zero.
Therefore we only need to compute the order of the integral As before, the largest term is obtained when we apply 1 It is enough to show that To finish the proof, we need to show that the latter integral is not equal to zero.
After integration by parts 2k times and the change of the variable t = z 2 the integral becomes Using integration by parts in the opposite order and observing that P (t) is the Taylor polynomial of (1 + t) −p/2 , we get dt.The latter is clearly a nonzero constant.Now we are ready to prove the main result of this section.
Theorem 3.4.For every 1 ≤ k < l < n − 3 there exists an origin-symmetric convex body K ∈ R n that does not belong to I k , but belongs to I l .
Proof.For a small ǫ > 0 define a body K by , where E is the ellipsoid with the norm .
Convexity of K follows along the lines of Lemma 2.3.(One will need ǫ n−k−3/2−2 to be small, which is the case since n − 3 > k).
Consider the −lth power of the norm of K.