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 \ (1-o (1) ) d d2^{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 ₂ norm. Our main tools are the graph-theoretic result used in the ₂ construction and recent progress on the Bourgain slicing problem.