Dyadic N-Dimensional Polytope

• Dyadic N-dimensional polytope is a convex polytope defined by the intersection of a real polytope with Dⁿ, featuring vertices in dyadic rationals and closure under arithmetic mean.
• It is finitely generated by its vertices and lower-dimensional walls, exhibiting unique combinatorial and topological behaviors distinct from classical lattice polytopes.
• Its structured algebraic properties enable efficient computational algorithms for convexity analysis, bridging geometric intuition and symbolic representation.

A Dyadic N-Dimensional Polytope is a geometric and algebraic object formed by the intersection of an N-dimensional real convex polytope whose vertices all lie in the dyadic space Dⁿ, where D denotes the ring of dyadic rationals (i.e., rational numbers with denominators that are powers of two). Dyadic polytopes have unique closure properties under arithmetic mean, are always finitely generated as algebraic structures, and embody a diverse range of combinatorial and topological behaviors that distinguish them sharply from their real or integer lattice counterparts. They are of interest in convex algebra, combinatorial geometry, and computational applications where dyadic representations and finite generator sets are central.

1. Foundations: Dyadic Rationals and Affine Spaces

The dyadic rationals are defined as $D = \mathbb{Z}[1/2]$, encompassing all $p / 2^k$ with $p \in \mathbb{Z}$ and $k \in \mathbb{N}$. The affine space $D^n$ admits a binary operation of arithmetic mean: $x \circ y = \frac{x + y}{2}$ This operation is closed in $D^n$—a property not shared by general rational or integer spaces—which underpins the algebraic structure of dyadic affine spaces. Dyadic affine spaces considered with arithmetic mean form commutative, idempotent, and entropic groupoids (or CB-modes), making them ideal for convex and combinatorial constructions.

2. Geometric Construction and Definition

A dyadic n-dimensional convex set is any intersection $P = C \cap D^n$, where $C \subset \mathbb{R}^n$ is a convex set. A dyadic n-dimensional polytope is the specific case where $C$ is a polytope (a bounded convex set with flat faces) and all its vertices are in $D^n$. Thus,

$P = C \cap D^n$

with $C$ an n-polytope with dyadic vertices. Algebraically, $P$ can be described as the subalgebra of $(D^n, \circ)$ generated by its vertices and lower-dimensional "walls," reflecting the closure under arithmetic mean.

Relevant formulas include: $$P = \operatorname{conv}_D(P) = \operatorname{conv}_{\mathbb{R}}(P) \cap D^n$$ for dyadic convex hull, and for simplices,

$$S_n = \left\{ \sum_{i=0}^{n} x_i r_i : r_i \in D, \sum_{i=0}^n r_i = 1 \right\}$$

3. Algebraic Structure and Generation

A key result is that every dyadic polytope is finitely generated as a subgroupoid under arithmetic mean (Matczak et al., 23 Mar 2024). This finite generation can involve either the set of vertices alone (for dyadic simplices), or vertices plus generators from lower-dimensional walls for more complex polytopes. Only dyadic simplices are generated solely by their vertices. A general dyadic polytope $P$ requires, in principle, the union of generators from its maximal walls (i.e., $(n-1)$-faces) and an interior simplex.

In one dimension, dyadic intervals are generated by two or three points, depending on their structure. In dimension $n$, an $n$-simplex needs exactly $n+1$ generators, matching the vertex count.

For subgroupoids of dyadic convex sets, Theorem 5.1 of (Matczak et al., 23 Mar 2024) establishes that such a subgroupoid $D \subset (D^n, \circ)$ is finitely generated if and only if it is isomorphic to a semipolytope—defined as having the same vertices as its parent polytope $C$ and agreeing on their interiors: $$\operatorname{conv}_D(D) = C,\quad D \cap \operatorname{int}(C) = \operatorname{int}(C)$$

4. Combinatorial and Topological Properties

Dyadic polytopes encompass a rich combinatorial diversity, even in low dimensions. For example, there are infinitely many nonisomorphic dyadic intervals in one dimension, each characterized by the structure of its generator set and the closure properties under means. In higher dimensions, the combinatorial structure depends both on the arrangement of dyadic vertices and on the configuration of lower-dimensional walls and faces. The intersection with dyadic space may result in polytopes with fractal-like boundaries, especially for semipolytopes, which may omit some parts of the original polytope's boundary but agree on the interior.

From a topological perspective, dyadic polytopes inherit the connectivity of their real counterparts but their algebraic generator sets constrain their structure in ways that can be recursively analyzed—by induction on dimension and density arguments for generating sets (Matczak et al., 23 Mar 2024).

5. Algorithmic and Computational Aspects

Because of the finiteness of generators, dyadic polytopes are tractable for symbolic and computational representation. This is especially significant in contexts such as finite groupoid computation, combinatorial convexity algorithms, and computer algebra systems implementing convex hulls and face enumeration in restricted affine spaces. The closure under arithmetic mean facilitates efficient recursive algorithms for enumeration and generation.

Methods for determining the minimal generator set are constructive, relying on analysis of maximal walls and embedded simplices. For instance, dyadic polygons in the plane may require both boundary and interior generators, and counting such generators involves combinatorial investigation of the convex subset's intersection properties with $D^2$.

6. Connections and Applications

Dyadic N-dimensional polytopes serve as a bridge between algebraic and geometric approaches to convexity. Their paper is motivated by applications in combinatorial convexity, computational geometry, logic (subreducts of affine spaces), and the theory of convex groupoids. The results answer foundational questions, such as whether every dyadic polytope is generated by its vertices—a question resolved negatively except for simplices (Matczak et al., 23 Mar 2024)—and provide a general framework for determining the minimal generator requirements for a broad class of convex sets.

Notably, the algebraic characterization of semipolytopes opens new approaches to understanding which convex sets in dyadic affine space can be efficiently constructed from finite data, and the structure of their generator sets is intimately linked to their topological and combinatorial properties.

7. Summary of Key Formulas and Definitions

Concept	Formal Definition	Reference
Dyadic rationals	$D = \mathbb{Z}[1/2]$, numbers of the form $p/2^k$	(Matczak et al., 23 Mar 2024)
Arithmetic mean	$x \circ y = (x + y)/2$	(Matczak et al., 23 Mar 2024)
Dyadic polytope	$P = C \cap D^n$, $C$ real polytope with dyadic vertices	(Matczak et al., 23 Mar 2024)
Dyadic simplex	$$S_n = \{\sum x_i r_i \mid r_i \in D, \sum r_i = 1\}$$	(Matczak et al., 23 Mar 2024)
Semipolytope	$D$ subgroupoid: $conv_D(D) = C$, $D \cap int(C) = int(C)$	(Matczak et al., 23 Mar 2024)

Dyadic N-dimensional polytopes are thus characterized by their intersection properties, algebraic closure under dyadic mean, finite generation, and the notable classification that only simplices are vertex-generated within their affine dyadic space. This structure fundamentally enriches the theory of convexity in discrete affine settings and forms a foundation for further algebraic and combinatorial investigations.