Dimension-free estimates for covering functionals of simplices and ${\mathrm{\ell }}_p$ balls

We study ${\mathrm{\Gamma }}_{2^n}(K)$, the least positive number $\gamma >0$ such that an $n$-dimensional convex body $K$ can be covered by $2^n$ translates of $\gamma K$. For $n$-simplices ${\mathrm{\Delta }}_n$, we prove that ${\mathrm{\Gamma }}_{2^n}({\mathrm{\Delta }}_n)$, as a sequence in $n$, tends to $1\mathrm{/}2$. For the cross-polytope $B_1^n$, we show that ${\mathrm{\Gamma }}_{2^n}(B_1^n)\le 5\mathrm{/}6$ holds for all $n\ge 2$, and that ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}{\mathrm{\Gamma }}_{2^n}(B_1^n)\le 0.641\cdots$. Finally, we prove the existence of a constant ${\kappa }_{\ast }<1$ such that ${\mathrm{\Gamma }}_{2^n}(B_p^n)\le {\kappa }_{\ast }$ for all $n\ge 2$ and all $p\in [1,\mathrm{\infty }]$.