Raja's covering index of $L_p$ spaces

We study Raja’s covering index ${\mathrm{\Theta }}_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving $${\mathrm{\Theta }}_H(n)=n^{-1\mathrm{/}2}(n\in \mathbb{N});$$ in particular ${\mathrm{\Theta }}_H(2)=1\mathrm{/}\sqrt{2}$, thus answering a question of Raja about the precise two-piece covering index of $\text{elltwo}$. For scalar-valued Lebesgue spaces $L_p(\mu )$, $1\le p<\mathrm{\infty }$, we construct an explicit block decomposition of the unit ball yielding the upper bound ${\mathrm{\Theta }}_{L_p(\mu )}(n)\le n^{-1\mathrm{/}p}$ for all $n\in \mathbb{N}$; in particular ${\mathrm{\Theta }}_{{\mathrm{\ell }}_p}(n)\le n^{-1\mathrm{/}p}$. For $1<p<\mathrm{\infty }$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja’s general lower bound to give the sharp asymptotic estimate ${\mathrm{\Theta }}_{L_p(\mu )}(n)\asymp n^{-1\mathrm{/}p}$. We also obtain uniform upper bounds ${\mathrm{\Theta }}_{L_p(\mu ;E)}(n)\le n^{-1\mathrm{/}p}$ for Bochner spaces $L_p(\mu ;E)$ over non-atomic $\sigma$-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 ${\mathrm{\Theta }}_{L_p(M,\tau )}(n)\gtrsim n^{-1\mathrm{/}r}$ for non-commutative $L_p(M,\tau )$ 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.