New dense superball packings in three dimensions

Frank Vallentin; Maria Dostert

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New dense superball packings in three dimensions

Maria Dostert, Frank Vallentin

Metric Geometry · Submitted 28 Jun 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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 $p \in (1, 1.58]$.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,019 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.74 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.86 · mentions code or data availability 0.02.

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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 $p \in (1, 1.58]$.	contribution 0.99	0.96	0.65	asserted 1.00
s1	We conjecture that the family also exists for $p \in (1.58, \log_2 3 = 1.5849625\ldots]$.	future work 0.75	0.34	0.10	speculative 1.00
s2	Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.	result 0.97	0.75	0.15	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.93.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original 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.

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1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

See [ref] for studies of $ℓ_p$-ball packings in $^3$.

Exponential improvements for superball packing upper bounds · Introduction · 0.99

What it cites · 4 sentences without a confident label

Contests 1 paper

The density we obtained for $p = 1.4$ coincides with the value reported in [ref] , but the basis $e_1, e_2, e_3$ given in [ref] does not give a packing lattice since the $ℓ^p_3$-norm of $2e_1-e_2-e_3$ is too small.

Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra · First regime · contest 0.89

Uses the method or tool of 2 papers

We will apply the following theorem of Cohn, Kumar, and Minton [ref] , which is an effective implicit function theorem.

Optimal simplices and codes in projective spaces · A new family of lattice packings · 0.96
Minkowski [ref] (see also [ref] , [ref] , [ref] ) showed that after performing a suitable $_n(ℤ)$-transformation to $B$ we can reduce to the following three cases.

Densest Lattice Packings of 3-Polytopes · Determination of lattice packings by local optimization · 0.89

Compares with 1 paper

The density we obained for $p = 1.7$ coincides with the value reported in [ref] , but in [ref] a corresponding matrix is not given.

Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra · Second regime · 0.98

Cites as background 1 paper

Jiao, Stillinger, and Torquato [ref] determined four families of dense superball packings via computer simulations.

Optimal Packings of Superballs · Introduction · 0.99

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