Packings in classical Banach spaces

We obtain several new results on the simultaneous packing and covering constant $γ(\mathcal{X})$ of a Banach space $\mathcal{X}$, and its lattice counterpart $γ^(\mathcal{X})$. These constants measure how efficient a (lattice) packing by unit balls in $\mathcal{X}$ can be, the optimal case being that $γ(\mathcal{X})= 1$ and the worst that $γ(\mathcal{X})= 2$. Our first main result is that $γ(\mathcal{X})> 1$ whenever $B_\mathcal{X}$ admits a LUR point, which leads us to a negative answer to a question of Swanepoel. We also develop general methods to compute these constants for a large class of spaces. As a sample of our findings: (i) $γ^(\mathcal{X})= 1$ when $\mathcal{X}$ is a separable octahedral Banach space, or $\mathcal{X}= \mathcal{C}(\mathcal{K})$, where $\mathcal{K}$ is zero-dimensional; (ii) $γ(\ell_p(κ)\oplus_r \mathcal{X})= γ^(\ell_p(κ)\oplus_r \mathcal{X})= \frac{2}{2^{1/p}}$, whenever $\rm{dens}(\mathcal{X})< κ$ and $1\leq r\leq p< \infty$; (iii) $γ(L_p(μ))= γ^(L_p(μ))= \frac{2}{2^{1/p}}$ for $1\leq p\leq 2$ and every measure $μ$; (iv) there exist reflexive (resp. octahedral) Banach spaces $\mathcal{X}$ with $γ(\mathcal{X})= 2$. We leave a large area open for further research and we indicate several possible directions.

💡 Research Summary

The paper investigates two geometric constants associated with a Banach space X: the simultaneous packing and covering constant γ(X) and its lattice analogue γ⁎(X). These constants measure how efficiently one can pack unit balls in X so that a slight inflation of the balls yields a covering of the whole space. The optimal value is 1 (perfect tiling) and the worst possible value is 2.

The authors begin by reviewing known facts: for any infinite‑dimensional Banach space one always has 1 ≤ 2K(X) ≤ γ(X) ≤ γ⁎(X) ≤ 2, where K(X) denotes the Kottman constant. Earlier work had shown γ⁎(X) ≤ 3 (Rogers) and later improved to γ⁎(X) ≤ 2. A question raised by Swanepoel asked whether γ(X) always equals 2K(X). The present work provides a negative answer and, more generally, a systematic study of γ and γ⁎ for a wide variety of classical spaces.

The first major result (Theorem A) proves that if the unit ball B_X possesses a locally uniformly rotund (LUR) point, then γ(X) > 1. Consequently every infinite‑dimensional Banach space is isomorphic to a space Y with γ(Y) > 1 while still having K(Y)=2. This shows that the presence of an LUR point prevents a perfect tiling by unit balls, extending earlier observations that such spaces cannot be tiled at all.

The second part of the paper develops two general techniques for estimating γ and γ⁎.

1. Construction of discrete subgroups (Section 4) generalizes earlier lattice constructions. Using these, the authors prove that γ⁎(X)=1 for every separable octahedral Banach space and for C(K) when K is zero‑dimensional (Theorems 4.2 and 4.4). Thus, in many classical settings a lattice packing can be made to cover the space without any inflation.
2. ϕ‑octahedrality (Section 5) introduces a new geometric notion parameterized by a convexity modulus ϕ. A space is ϕ‑octahedral if, roughly, in every direction the norm grows at least as fast as prescribed by ϕ. This framework yields a lower bound γ(X) ≥ 2K(X)·(1+ϕ‑modulus) (Theorem 5.10), which refines the earlier inequality γ(X) ≥ 2K(X). Uniformly convex spaces turn out to be ϕ‑octahedral, allowing precise calculations for L_p‑spaces with 1 ≤ p ≤ 2.

Using these tools the authors obtain a series of concrete results (Theorem B):

• (I) γ⁎(X)=1 for separable octahedral spaces and for C(K) with K zero‑dimensional.
• (II) For any infinite cardinal κ, any Banach space X with density less than κ, and 1 ≤ r ≤ p < ∞, the direct sum ℓ_p(κ)⊕_r X satisfies γ=γ⁎=2·2^{‑1/p}. This extends earlier calculations for ℓ_p(κ) alone.
• (III) For any measure space (M,Σ,μ) and X with density smaller than that of L_p(μ), the sum L_p(μ)⊕_r X has γ and γ⁎ bounded between min{2·2^{‑1/p}, 2·2^{‑1/q}} and 2·2^{‑1/p} (q is the conjugate exponent). When p ≤ 2 the lower bound coincides with the upper bound, giving exact values γ=γ⁎=2·2^{‑1/p}.
• (IV) For super‑reflexive spaces the authors relate γ and γ⁎ to the modulus of convexity δ_X and the tangential modulus φ_X, obtaining explicit inequalities.
• (V) They construct Banach spaces (of density ω₁) with γ(X)=2; these can be chosen reflexive or octahedral. Hence the maximal possible value 2 is attained even in “nice’’ spaces, providing strong counterexamples to Swanepoel’s conjecture.

The paper also discusses several open problems: whether one can find a renorming of ℓ₂ with γ⁎>√2, whether super‑reflexive spaces can have γ=2, and whether uniformly convex spaces always admit lattice packings with γ⁎=1.

Overall, the work significantly advances the understanding of simultaneous packing and covering constants in infinite‑dimensional Banach spaces. By introducing LUR‑point arguments and the novel ϕ‑octahedral framework, the authors obtain exact values for many classical spaces (ℓ_p, L_p, C(K)) and exhibit a rich variety of behaviors—including spaces where γ attains its maximal value—thereby answering longstanding questions and opening new directions for research.