  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 $\ell_2$ norm. Our
main tools are the graph-theoretic result used in the $\ell_2$ construction and
recent progress on the Bourgain slicing problem.
