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) $\mathcal I_l \not\subset \mathcal I_k$, $1\le k<l < n-3$.

💡 Research Summary

The paper investigates strict inclusion relations among classes of convex bodies defined via Koldobsky’s notion of k‑intersection bodies. For a given integer k (1 ≤ k < n) the class Iₖ consists of all origin‑symmetric convex bodies that are k‑intersection bodies, i.e., bodies whose radial function raised to the power –k has a Fourier transform that is a positive distribution. A second family, denoted Iₖᵐ, contains those convex bodies whose every m‑dimensional central section is a k‑intersection body. The author addresses two natural questions: (1) whether the hierarchy Iₖᵐ ⊂ Iₖ^{m+1} is strict, and (2) whether the hierarchy Iₖ ⊂ I_l (for k < l) is strict.

The first main result proves that for any integers k, m with k + 3 ≤ m < n, there exists an origin‑symmetric convex body K that belongs to Iₖᵐ but not to Iₖ^{m+1}. The construction starts with the Euclidean unit ball and perturbs its radial function by a small parameter ε>0 using an ellipsoid E whose semi‑axes are 1 in the first m coordinates and ε in the remaining n‑m coordinates. Explicitly, the perturbed norm satisfies
‖x‖_k^{‑k}=|x|^{‑k} − 2 ε^{m‑k}‖x‖_k^{‑k} E.
For sufficiently small ε the body remains convex (Lemma 2.3). By restricting to any m‑dimensional subspace H, the section K∩H has the same functional form, and a direct Fourier‑transform computation shows that its transform is non‑negative, hence each m‑section is a k‑intersection body (Lemma 2.4). However, selecting a specific (m+1)‑dimensional subspace H₀, the Fourier transform evaluated at a suitable direction becomes negative, demonstrating that K∩H₀ fails to be a k‑intersection body (Lemma 2.5). Consequently Iₖᵐ ⊈ Iₖ^{m+1}.

The second main result establishes that for any 1 ≤ k < l < n‑3 there exists a convex body that is an l‑intersection body but not a k‑intersection body, i.e., I_l ⊈ I_k. The proof relies on a refined analysis of Fourier transforms of homogeneous functions of the form |x|^{‑p} multiplied by a perturbed ellipsoidal factor. Lemma 3.1 provides explicit integral representations for the Fourier transform of f(θ) r^{‑p} depending on the parity of the exponent, involving spherical Laplacians and logarithmic terms. Lemma 3.2 supplies a spherical Parseval identity linking integrals of products of such transforms to integrals of the original functions. Lemma 3.3 then estimates the size of the Fourier transform of the perturbed term, showing that it behaves like ε^{‑(n‑p‑q‑1)} (up to logarithmic corrections) as ε→0. By choosing p and q appropriately (with p<q) and letting ε tend to zero, the sign of the Fourier transform can be forced to change, which means the perturbed body is not a k‑intersection body while still satisfying the positivity condition for the larger exponent l.

Throughout, the author exploits the linearity of the Fourier transform, the homogeneity of the radial functions, and the self‑adjointness of the spherical Laplacian. The perturbation technique with a small ε allows simultaneous control of convexity (via curvature estimates) and of the sign of the Fourier transform. This method extends earlier constructions by Schlieper, Borwein, Kalton‑Koldobsky, and others, which dealt with embeddings of Banach spaces into L_p for negative p. The present work answers affirmatively the open question posed by Koldobsky: the inclusions among the families Iₖ and Iₖᵐ are indeed strict, and the hierarchy does not collapse for intermediate values of k and m.

In summary, the paper provides explicit convex bodies that separate the classes Iₖᵐ and Iₖ^{m+1} as well as I_k and I_l, using Fourier‑analytic criteria, careful perturbations by anisotropic ellipsoids, and detailed asymptotic estimates. These results deepen our understanding of the geometry of intersection bodies and their connections to embeddings of normed spaces into negative‑exponent L_p spaces.