New dense superball packings in three dimensions

TL;DR

This paper introduces a new family of dense lattice packings for superballs in three dimensions, expanding understanding of optimal arrangements for these shapes within a specific p-norm range.

Contribution

The authors construct a novel family of lattice packings for superballs in 3D and conjecture its extension beyond the proven p-range, advancing geometric packing theory.

Findings

New lattice packings for superballs with p in (1, 1.58]

Each superball has 14 neighbors in the packing

Conjecture extends the family for p up to approximately 1.585

Abstract

In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p \in (1, 1.58]$. We conjecture that the family also exists for $p \in (1.58, \log_2 3 = 1.5849625\ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.