Raja's covering index of $L_p$ spaces
Tomasz Kania, Natalia Maślany
Abstract
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.