Simultaneous packing and covering in sequence spaces

We adapt a construction of Klee (1981) to find a packing of unit balls in ${\mathrm{\ell }}_p$ ($1\le p<\mathrm{\infty }$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1\mathrm{/}p}$ covers the whole space. We show that the value $2^{1-1\mathrm{/}p}$ is optimal.