Lower bounds for sphere packing in arbitrary norms

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least $$\frac{(1-o(1))d\log d}{2^{d+1}},$$ improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of Campos, Jenssen, Michelen, and Sahasrabudhe in the ${\mathrm{\ell }}_2$ norm. Our main tools are the graph-theoretic result used in the ${\mathrm{\ell }}_2$ construction and volume bounds from convex geometry due to Petty and Schmuckenschläger.