[Paper Review] Perfect but not generating Delaunay polytopes

Mathieu Dutour Sikirić, Konstantin Rybnikov|ArXiv.org|May 28, 2009

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper constructs an infinite family of $n$-dimensional perfect Delaunay polytopes $P(n,s)$ for $s \geq 2$ and $n+1 \geq 4s$, which are perfect but not generating. It shows that for $6s < n+1$ (odd $n$) or $6s < n$ (even $n$), $P(n,s)$ remains a perfect Delaunay polytope in a strictly larger lattice $V^4_{n,s}$, providing the first known example of a perfect Delaunay polytope that is not generating, analogous to the classical $\mathsf{A}_9$ and $\mathsf{A}_9^2$ relationship in the homogeneous case.

ABSTRACT

In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is that of a lattice with a perfect Delaunay polytope: the vertices of a perfect Delaunay polytope are the analogs of minimal vectors in a perfect lattice. We find a new infinite series $P(n,s)$ for $s\geq 2$ and $n+1\geq 4s$ of $n$-dimensional perfect Delaunay polytopes. A remarkable property of this series is that for certain values of $s$ and all $n \ge 13$ one can add points to the integer affine span of $P(n,s)$ in such a way that $P(n,s)$ remains a perfect Delaunay polytope in the new lattice. Thus, we have constructed an inhomogeneous analog of the remarkable relationship between $\sfA_9$ and $\sfA_9^2$.

Motivation & Objective

• To construct an infinite family of perfect Delaunay polytopes that are not generating, addressing a gap in the classification of perfect Delaunay polytopes.
• To establish an inhomogeneous analog of the classical $\mathsf{A}_9 \subset \mathsf{A}_9^2$ relationship, where a perfect lattice admits a larger lattice preserving the same perfect Delaunay polytope.
• To demonstrate that perfect Delaunay polytopes can exist in lattices where their vertex span does not generate the full lattice, challenging the assumption that all perfect Delaunay polytopes are generating.
• To provide a systematic construction of such polytopes using symmetric subsets of the $\{0,1\}^{n+1}$ hypercube and lattice extensions via $V^2_{n,s}$ and $V^4_{n,s}$.

Proposed method

• Define $P(n,s)$ as the convex hull of $V_{n,s} \cup (t_{n,s} + V_{n,s})$, where $V_{n,s}$ is the set of $\{0,1\}^{n+1}$ vectors with exactly $s$ ones and $t_{n,s}$ is a specific shift vector.
• Construct the lattice $V^2_{n,s} = V_{n,s} \cup (t_{n,s} + V_{n,s})$, which is shown to be a full-rank lattice and affinely generated by $P(n,s)$.
• Prove that $P(n,s)$ is perfect by showing that the unique positive definite quadratic form (up to scaling) realizing its Delaunay property is $q_{n,s}(x) = 2\sum_{i=0}^{4s-1}x_i^2 + \sum_{i=4s}^{n}x_i^2$, derived from the center $v_{n,s}$ of the circumscribed sphere.
• Define $V^4_{n,s} = V^2_{n,s} \cup (w_{n,s} + V^2_{n,s})$, where $w_{n,s}$ is a carefully chosen vector to extend the lattice while preserving the Delaunay property of $P(n,s)$.
• Use the closest vector problem in $V^4_{n,s}$ to show that the squared distance from $v_{n,s}$ to points in $w_{n,s} + V^2_{n,s}$ exceeds the squared radius $\frac{3s}{2}$ of the circumscribed sphere, ensuring $P(n,s)$ remains a Delaunay polytope.
• Establish the condition $6s < n+1$ (odd $n$) or $6s < n$ (even $n$) as sufficient for $P(n,s)$ to remain Delaunay in $V^4_{n,s}$, using lower bounds on $q_{n,s}$-norms of displacement vectors.

Experimental results

Research questions

• RQ1 Can perfect Delaunay polytopes exist that are not generating, i.e., whose integer affine hull does not span the ambient lattice?
• RQ2 Is there an inhomogeneous analog of the $\mathsf{A}_9 \subset \mathsf{A}_9^2$ phenomenon, where a perfect lattice admits a larger lattice preserving the same perfect Delaunay polytope?
• RQ3 What conditions on $n$ and $s$ ensure that $P(n,s)$ remains a Delaunay polytope in an extended lattice $V^4_{n,s}$?
• RQ4 How can symmetric subsets of the hypercube $\{0,1\}^{n+1}$ be used to construct perfect Delaunay polytopes with controlled symmetry and lattice structure?

Key findings

• The polytope $P(n,s)$ is perfect for all $s \geq 2$ and $4s \leq n+1$, with a unique positive definite quadratic form $q_{n,s}$ up to scaling, given by $2\sum_{i=0}^{4s-1}x_i^2 + \sum_{i=4s}^{n}x_i^2$.
• For $n \geq 13$ and $s=2$, $P(13,2)$ is the first known example of a perfect Delaunay polytope that is not generating, as $L(P) \neq L'$ in the extended lattice $L' = V^4_{13,2}$.
• The condition $6s < n+1$ (odd $n$) or $6s < n$ (even $n$) ensures that $P(n,s)$ remains a Delaunay polytope in the extended lattice $V^4_{n,s}$, as the closest vectors in the new coset are farther than the circumscribed sphere radius.
• The squared radius of the circumscribed sphere of $P(n,s)$ is $\frac{3s}{2}$, and the center is $v_{n,s} = \left(\frac{1}{4},\dots,\frac{1}{4}, \frac{1}{2},\dots,\frac{1}{2}\right)$ with $4s$ entries of $\frac{1}{4}$ and $n+1-4s$ of $\frac{1}{2}$.
• The construction generalizes known examples: $P(7,2)$ is the Gosset polytope $3_{21}$, and $P(8,2)$ matches the Delaunay polytope $D^8_2$ from [DER07].
• The lattice $V^4_{n,s}$ is an index-2 extension of $V^2_{n,s}$, and $P(n,s)$ remains Delaunay in $V^4_{n,s}$ because the minimal $q_{n,s}$-norms of vectors in $w_{n,s} + V^2_{n,s}$ exceed $\frac{3s}{2}$ under the stated conditions.