Lattice 3-polytopes with few lattice points

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proposition_2_2: Two ordered d-dimensional sets of n lattice points with the same volume vector are related by a unique affine map of determinant one respecting the order, and that map is a unimodular equivalence when the gcd of the volume vector's entries is 1.

theorem_1_1: Blanco and Santos's classification: every lattice 3-polytope with exactly five lattice points is unimodularly equivalent to exactly one entry of their Table 1, which has four width-one types (two isolated classes and two infinite families) and nine classes of width two, none of larger width.

theorem_1_2: Blanco and Santos's structure theorem for a lattice 3-polytope with five lattice points: signatures (2,2), (2,1) and (3,2) force width one, and in signatures (3,1) and (4,1) an affine integer functional takes the values 1, 1, 0, 0, h on the lattice points with h equal to -1 or -2.

theorem_1_3: Blanco and Santos's finiteness theorem: for each n >= 4 there are only finitely many unimodular equivalence classes of lattice 3-polytopes with exactly n lattice points and width greater than one.

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The copy read for this card carries the stamp "arXiv:1409.6701v3 [math.CO] 12 May 2016" and prints no notice, and the journal edition was not compared; the arXiv abstract page names arXiv's non-exclusive distribution license (https://arxiv.org/abs/1409.6701v3, read 2026-10-02), every other right reserved.

Mónica Blanco, Francisco Santos, "Lattice 3-polytopes with few lattice points," SIAM J. Discrete Math. 30 (2016), no. 2, 669--686, DOI 10.1137/15M1014450.

Overview

Question and scope. Blanco and Santos classify, up to affine unimodular equivalence, all three-dimensional lattice polytopes having exactly five lattice points, and establish a finiteness theorem for fixed-size lattice $3$-polytopes of width greater than one. The copy read for this card is arXiv:1409.6701v3 (12 May 2016), 19 pages; the journal pagination 669–686 was not used, and the precise locators below are theorem, proposition, equation, table, and section numbers.

Main classification. Theorem 1.1 and Table 1 give a complete, irredundant list. The width-one part consists of the four types in Theorem 4.1: two isolated classes of signatures $(2,2)$ and $(3,1)$; a signature-$(2,1)$ family parametrized by $0\le p\le\lfloor q/2\rfloor$ with $\gcd(p,q)=1$; and a signature-$(3,2)$ family parametrized by $0<a\le b$ with $\gcd(a,b)=1$. Outside width one there are exactly nine classes, all of width two, and none of larger width. Their volume vectors are

• $(-9,3,3,3,0)$ in signature $(3,1)$ (Theorem 4.2);
• $(-5,1,1,1,2)$, $(-7,1,1,2,3)$, $(-11,1,3,2,5)$, $(-13,3,4,1,5)$, $(-17,3,5,2,7)$, and $(-19,5,4,3,7)$ among the nonsymmetric signature-$(4,1)$ cases (Theorem 4.3);
• $(-4,1,1,1,1)$ and $(-20,5,5,5,5)$ in the symmetric signature-$(4,1)$ cases (Theorem 4.4).

The structural input is Theorem 1.2, proved as Theorems 3.3 and 3.4. Theorem 3.3 states that signatures $(3,2)$, $(2,2)$, and $(2,1)$ force width one. Theorem 3.4 states that for signature $(3,1)$ or $(4,1)$, after choosing a largest-volume empty tetrahedron, an integer affine functional takes values $(1,1,0,0,h)$ on the five points with $h\in\{-1,-2\}$. In signature $(4,1)$, the case $h=-2$ is exactly the symmetric volume-vector case $(-4q,q,q,q,q)$.

Invariants and method. Definition 2.1 and Equation (1) define the volume vector by signed normalized determinants. Equation (2) identifies the five entries for a five-point configuration with the coefficients of its unique affine dependence. Proposition 2.2 shows that equal ordered volume vectors determine a unique determinant-one affine map; when the gcd of all entries is one, this map is a $\mathbb Z$-equivalence. Empty tetrahedra are normalized using White’s cited classification, Theorem 2.4, while Lemma 2.5 supplies three explicit congruence-and-coprimality tests for emptiness. Lemma 2.6 permits any selected vertex of an empty tetrahedron to be moved to the origin, possibly replacing $p$ by $p^{-1}\pmod q$.

Proposition 3.1 sends White’s tetrahedron $T(p,q)$ to the standard simplex $T_0$ while replacing $\mathbb Z^3$ by the finer lattice $\Lambda(p,q)=\langle(1/q,-1/q,p/q)\rangle+\mathbb Z^3$. Lemma 3.2 controls lattice points in two distinguished triangles inside a fundamental rectangle of this lattice. In Theorem 3.3, the fifth point is forced into $2T_0$; the rectangle lemma either produces an extra lattice point or another width-one functional. In Theorem 3.4, maximality of the selected empty tetrahedron puts the fifth point in $[-1,0]^3$, yielding the three-level functional. Section 4 then performs explicit planar case analyses. Theorems 4.2–4.4 reduce the admissible parameters by emptiness, coprimality, and lattice-point exclusion; Equations (3) and (4) encode the central inequalities in the nonsymmetric $(4,1)$ analysis.

General finiteness and limitations. Corollary 5.1, restated as Theorem 1.3, proves that for every fixed $n\ge4$ there are only finitely many equivalence classes of size-$n$ lattice $3$-polytopes of width greater than one. Its proof combines three explicitly cited external results—Hensley’s volume bound for a fixed number of interior lattice points, Lagarias–Ziegler finiteness at bounded volume, and Nill–Ziegler’s projection theorem for hollow polytopes—with a direct treatment of polytopes projecting onto twice a unimodular triangle. Section 5 only sketches a recursive classification scheme using minimal and quasi-minimal polytopes (Definition 5.4); Proposition 5.5 shows that infinitely many minimal $3$-polytopes exist. The large-size structure theorem stated as Theorem 5.6 is attributed to the subsequent paper [6], not proved here. Thus the paper completely classifies size five and proves fixed-size finiteness beyond width one, but it does not classify arbitrary sizes or provide quantitative enumeration bounds.

Relation to E963

Write

$$\delta(A)=\max\{|D|:D\subseteq A,\ \sum_{d\in D}\eta_d d\text{ is distinct for all }\eta\in\{0,1\}^D\}.$$

Equivalently, $D$ is dissociated precisely when

$$\sum_{d\in D}\varepsilon_d d=0,\qquad \varepsilon_d\in\{-1,0,1\},$$

forces every $\varepsilon_d=0$. In E963’s notation, $f(n)=\min_{|A|=n}\delta(A)$ for $A\subset\mathbb R$.

The paper’s closest notion is a distinct-pair-sums (dps) lattice polytope, discussed in the final paragraph of Section 1. For five lattice points in dimension three, it states that dps is equivalent to the signature being neither $(2,2)$ nor $(2,1)$. This is not the same as dissociation: dps also forbids a repeated-summand equality $2x_i=x_j+x_k$, whose coefficient $2$ is outside $\{-1,0,1\}$, while it does not test collisions between subset sums of arbitrary cardinalities.

This connection is limited. E963 concerns arbitrary finite subsets of $\mathbb R$, with no convex-hull saturation, fixed affine rank, or width hypothesis. Moreover, dissociation is not invariant under translation, whereas the paper classifies configurations up to affine unimodular maps that include translations. The parity observation in Section 1—that a dps lattice $d$-polytope has at most $2^d$ lattice points—resembles E963’s logarithmic scale, but it is an upper bound for an entire dps configuration in a prescribed lattice dimension, not a theorem extracting a dissociated subset from an arbitrary real set. Neither Theorem 1.1 nor Corollary 5.1 supplies a bound on $f(n)$, and the paper does not prove or refute $f(n)\ge\lfloor\log_2 n\rfloor$.

Bears on.

• #963: the paper does not mention dissociated sets. The note above shows that Theorem 1.1 with Proposition 2.2 decides which five-point lattice configurations of affine rank three, with no further lattice points in their convex hull, become dissociated sets of reals under a generic affine embedding: all but the signature-$(2,2)$ class. It gives no bound on $f(n)$ and does not answer the question.

Results.

• Theorem 1.1 (p. 1): Every lattice 3-polytope of size 5 is unimodularly equivalent to exactly one entry of Table 1 (p. 2): four width-one types and nine classes of width two, none wider.
• Theorem 1.2 (p. 2): Signatures (2,2), (2,1), (3,2) force width one; in signatures (3,1), (4,1) an affine integer functional takes values (1,1,0,0,h), h in {-1,-2} (Theorems 3.3, p. 8, and 3.4, p. 10).
• Theorem 1.3 (p. 3): For each n >= 4 there are finitely many lattice 3-polytopes of size n and width greater than one (Corollary 5.1, p. 17).
• Proposition 2.2 (p. 4): Equal volume vectors give a unique determinant-one affine map between the ordered sets, a unimodular equivalence when the entries have gcd 1.

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