Lower bounds for sphere packing in arbitrary norms
Carl Schildkraut
TL;DR
This work proves a universal lower bound for translational sphere packings in any d-dimensional normed space: for every centrally symmetric convex body K in \mathbb{R}^d, the translational packing density satisfies delta_T(K) >= (1-o(1)) \frac{d \log d}{2^{d+1}} as d grows. The authors adapt the amorphous-packing framework used for the Euclidean case to arbitrary norms by combining a Poisson-sampled point set with a graph G(X,K) of intersecting translates, a pruning step to control degree and codegree, and a CJMS-style independent-set bound to extract centers of a packing; a crucial geometric input is a bound on vol(I_K), the set of vectors causing large intersections between translates, which is obtained via Schmuckenschläger’s link between K, its polar projection body \Pi^*K, and Petty’s inequality. The result extends the CJMS improvement (logarithmic gain in d) from the Euclidean norm to all norms, and yields corollaries for non-symmetric bodies via Rogers–Shephard, while highlighting the role of convex-geometry volume bounds and log-concavity arguments. Overall, the paper significantly strengthens translational packing lower bounds in high dimensions across arbitrary norms and connects amorphous packings with projection-geometry tools.
Abstract
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 volume bounds from convex geometry due to Petty and Schmuckenschläger.