Dyadic N-D Convex Sets

• Dyadic N-dimensional convex sets are intersections of real convex sets with dyadic rational grids, capturing unique algebraic and geometric properties.
• They leverage the arithmetic mean operation to form commutative binary modes, enabling classification via affine automorphisms and ensuring finite generation.
• Their effective computation in discrete geometry and combinatorics supports applications in system verification, digital geometry, and abstract interpretation.

A dyadic N-dimensional convex set is defined as the intersection with N-dimensional dyadic space $D^N$ of an N-dimensional real convex set, where $D$ denotes the dyadic rationals (rationals whose denominator is a power of $2$). Such sets arise naturally in discrete geometry, algebraic combinatorics, and universal algebra, bridging continuous convex geometry and algebraic structures defined over the dyadic numbers. The study of these sets leverages the arithmetic mean operation as a central structural tool and reveals important distinctions from classical real convex sets due to the underlying dyadic lattice.

1. Formal Definition and Algebraic Structure

A dyadic N-dimensional convex set $P$ is given by

$P = C \cap D^N$

where $C \subseteq \mathbb{R}^N$ is a convex set, and $D^N$ is the N-dimensional dyadic grid. If $C$ is a polytope whose vertices lie in $D^N$, then $P$ is referred to as a dyadic N-dimensional polytope (Matczak et al., 2024, Mućka et al., 7 Oct 2025). This structure is equivalently described algebraically as a subalgebra of a faithful affine space over the ring of dyadic numbers under the binary arithmetic mean operation: $x \circ y = \frac{x + y}{2}$ This operation is idempotent, commutative, and entropic (i.e. medial), making dyadic convex sets examples of commutative binary modes (CB-modes) (Czédli et al., 2012, Matczak et al., 2024, Mućka et al., 7 Oct 2025). Such algebraic objects satisfy, for all $x, y, z, t \in D^N$,

$$(x \circ y) \circ (z \circ t) = (x \circ z) \circ (y \circ t)$$

This property underlies many structural results and is crucial for universal algebraic approaches to convexity.

2. Generation and Minimal Generators

Dyadic polytopes are always finitely generated as subgroupoids under the arithmetic mean (Matczak et al., 2024). Specifically, a dyadic simplex (the dyadic analog of an $N$-simplex, with $N+1$ affinely independent vertices) is generated solely by its vertices—these points freely generate the entire simplex in the CB-mode sense. More general dyadic polytopes may require additional generators beyond the set of vertices; these are often extracted from the structure of the polytope's walls (lower-dimensional facets) and interior points. The minimal number of generators for a dyadic convex set can be determined via recursive decomposition—first generating all $(N-1)$-dimensional walls and then supplementing with generators from intrinsic simplices within the polytope (Matczak et al., 2024). The class of finitely generated dyadic convex sets coincides precisely with those isomorphic to semipolytopes, i.e., algebraically and geometrically well-behaved subgroupoids that share their vertex and interior structure with the full dyadic polytope.

3. Isomorphism and Affine Automorphisms

The algebraic isomorphism class of a dyadic convex set, considered as a CB-mode, is typically determined by affine automorphisms of the underlying dyadic affine space. For the dyadic setting ($F = D$), or more generally any subfield or suitable subring $T$ of $\mathbb{R}$, isomorphism between convex subsets (as barycentric algebras) holds if and only if there exists an affine automorphism $\psi \in \text{Aff}_{F}(F^N)$ mapping one set onto another (Czédli et al., 2012). For dyadic convex sets, these automorphisms are described by affine transformations with coefficients in $D$ (possibly matrices with dyadic entries). When considering only the algebraic structure, the necessary and sufficient condition for isomorphism is the preservation of the CB-mode via such affine maps. This result underpins classification schemes and enables the separation of isomorphism types within the space of dyadic convex sets (Mućka et al., 7 Oct 2025).

4. Classification in Lower Dimensions and Normal Forms

In dimension two, the study of dyadic polygons—especially dyadic triangles—offers a concrete illustration of the framework and normal forms (Mućka et al., 7 Oct 2025). Dyadic triangles can always be reduced, via dyadic affine transformations, to a normal form in which one vertex is fixed at the origin (the "pointed" vertex), and the other vertices have integer coordinates subject to parity and coprimality constraints. For example, any dyadic triangle is classified up to isomorphism by a triple $(i, j, m)$ of odd integers: $$T_{i,j,m}: \quad A = (0,0),\quad B = (i,j),\quad C = (m,0)$$ where the classification depends on the coprimality and relative sizes of $i, j, m$. The area formula

$P = \frac{1}{2}|\det([B-A, C-A])|$

shows that under dyadic automorphisms, areas of isomorphic triangles scale by powers of two. Right triangles and "hat" triangles are distinguished by further parity and boundary conditions; the methodology extends to higher dimensional simplices, where analogous encoding tuples and normal forms are expected (Mućka et al., 7 Oct 2025).

5. Polyhedrality, Convex Hull Algorithms, and Computability

The convex hull of a regular set of integer vectors encoded via dyadic representations (e.g., as strings of digit vectors) is always a polyhedral convex set and is effectively computable (0812.1951). For sets represented symbolically (Number Decision Diagrams, NDDs) over base-$r$ digit alphabets (including dyadic bases), the convex hull is characterized as

$$\text{conv}(X) = \{ x \in \mathbb{R}^N : (x,1) \in C(R) \},\quad C(R) = \left\{ \sum_{r \in R} t_r r : t_r \geq 0 \right\}$$

where $R$ is a finite set of rays. Recursive methods involving scaling and translation mappings (e.g., $\Gamma_\sigma(x) = r^{|\sigma|}x + \rho(\sigma)$ and normalization via $\xi(\sigma) = \rho(\sigma) / (1-r^{|\sigma|})$) allow explicit computation of convex hulls for sets defined by regular languages, with complexity exponential in the language description. This result is particularly relevant for system verification, over-approximation of infinite state spaces, and abstract interpretation (0812.1951).

6. Applications, Further Properties, and Interplay with Other Convexity Concepts

Dyadic N-dimensional convex sets are instrumental in areas requiring discrete convex structures compatible with algebraic operations—digital geometry, combinatorics, and computational geometry. Their study reveals key algebraic features (idempotency, mediality) and supports the classification of convex sets via algebraic invariants (Czédli et al., 2012). The geometric and algebraic properties diverge significantly from classical convex sets over $\mathbb{R}^N$; most notably, dyadic convex sets must be carefully distinguished by isomorphism type induced by the dyadic coordinate structure. In particular, generic polytope generation requires more than just extreme points except for simplices, reflecting a subtle interplay of combinatorics and geometry (Matczak et al., 2024).

Connections are made between dyadic convex sets and higher-level convexity concepts—including polyhedrality, extremal decomposition, and sectioning via hyperplanes—though dyadic sets are generally combinatorial in construction and lack the global smoothness and algebraic rigidity of quadric convex bodies (Soltan, 2010). This distinction is crucial when generalizing classification schema and invariants to multidimensional discrete settings.

7. Summary Table: Key Properties of Dyadic N-Dimensional Convex Sets

Feature	Description	Reference
Definition	$P = C \cap D^N$; intersection of real convex set with dyadic grid	(Matczak et al., 2024, Mućka et al., 7 Oct 2025)
Algebraic structure	CB-mode (commutative, entropic, idempotent) under arithmetic mean	(Czédli et al., 2012)
Isomorphism condition	Affine automorphism over $D^N$ preserves the barycentric algebra	(Czédli et al., 2012)
Polytope generation	Always finitely generated; only dyadic simplices by vertices alone	(Matczak et al., 2024)
Classification of triangles/simplices	Canonical encoding by integer tuples; pointed normal forms possible	(Mućka et al., 7 Oct 2025)
Convex hull computability	Polyhedral, effective algorithm over regular languages (NDDs)	(0812.1951)

In conclusion, dyadic N-dimensional convex sets unify discrete convexity, algebraic structure, and computational viability, exhibiting rich classification behaviors and distinctive generational and isomorphism properties traceable to their underlying dyadic algebraic framework. Their study reveals nuanced differences from real convex geometry, particularly in generation, isomorphism, and combinatorial decomposition, providing an essential foundation for both theoretical investigations and practical computation in discrete and digital settings.