Raja's covering index of $L_p$ spaces

We study Raja’s covering index $Θ_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving [ Θ_H(n)=n^{-1/2}\qquad(n\in\mathbb N); ] in particular $Θ_H(2)=1/\sqrt2$, thus answering a question of Raja about the precise two-piece covering index of $\elltwo$. For scalar-valued Lebesgue spaces $L_p(μ)$, $1\le p<\infty$, we construct an explicit block decomposition of the unit ball yielding the upper bound $Θ_{L_p(μ)}(n)\le n^{-1/p}$ for all $n\in\mathbb{N}$; in particular $Θ_{\ell_p}(n)\le n^{-1/p}$. For $1<p<\infty$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja’s general lower bound to give the sharp asymptotic estimate $Θ_{L_p(μ)}(n)\asymp n^{-1/p}$. We also obtain uniform upper bounds $Θ_{L_p(μ;E)}(n)\le n^{-1/p}$ for Bochner spaces $L_p(μ;E)$ over non-atomic $σ$-finite measure spaces, with constants independent of the Banach space $E$; this shows that, at the level of power-type upper estimates, the covering index decays at the same rate regardless of the asymptotic geometry of~$E$ and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds $Θ_{L_p(M,τ)}(n)\gtrsim n^{-1/r}$ for non-commutative $L_p(M,τ)$ spaces associated with semifinite von Neumann algebras, where $r=\min{p,2}$. We do not attempt to optimise the exponent or constants in the non-commutative setting.

💡 Research Summary

The paper investigates the covering index Θ_X(n), a quantitative invariant introduced by Raja, for classical L_p spaces, their Bochner extensions, and non‑commutative L_p spaces. The covering index measures how well the unit ball B_X can be covered by n closed convex sets, each of which contains a large “essential inradius” – a ball of maximal radius that lives in an infinite‑dimensional subspace of X. This invariant bridges local geometric properties (asymptotic uniform smoothness, AUS) and global combinatorial complexity (Szlenk index).

The authors first treat infinite‑dimensional Hilbert spaces H. By orthogonal decomposition H = ⊕{j=1}^n H_j into n infinite‑dimensional subspaces and defining A_j = {x∈B_H : ‖P_j x‖ ≤ n^{-1/2}} (P_j the orthogonal projection), they obtain a covering B_H = ⋃{j=1}^n A_j with essential inradius exactly n^{-1/2} for each A_j. The lower bound follows from Raja’s “goal derivation” applied to Hilbert balls: for any ε > n^{-1/2} the n‑th derived set