Multiary Algebraic B-Convexity

• The paper establishes a rigorous algebraic framework for ℬ-convexity using multiary, nonassociative, idempotent operations that generalize traditional convex hull constructions.
• It employs a determinant-like summation mechanism to define convex combinations and obtain separation via idempotent linear forms over ℝⁿ.
• The framework yields new results on external polytope representations through halfspace intersections, extending classical max-plus convexity to multi-point settings.

The multiary algebraic form of ℬ-convexity denotes a convexity structure grounded in the Kuratowski–Painlevé limit of parametrized polytopes, formalized by a nonassociative, idempotent, symmetric algebraic operation. The recent work (Briec, 18 Sep 2025) constructs an explicit algebraic framework extending earlier ad hoc topological definitions, yielding new results on polytope structure, separation, and representability within ℝⁿ.

1. Algebraic Formulation of Multiary ℬ-Convexity

The foundational concept is the ℬ-convex hull generated by limiting procedures over finite sets. For $A = \{x^{(1)}, ..., x^{(m)}\} \subset \mathbb{R}^n$, one defines for each $p$ a convex hull via a one-parameter family of "ϕₚ-sums" (with $ϕₚ(\lambda) = \lambda^{2p+1}$), such that: $$Co^{(p)}(A) = \left\{ \;ϕ_p^{-1}\left[ \sum_{j=1}^m t_j^{2p+1} ϕ_p(x^{(j)}) \right] : t_j \geq 0, \sum_j t_j = 1 \;\right\}$$ The multiary ℬ-convex hull is then the Kuratowski upper limit as $p \to \infty$: $Co^{(\infty)}(A) = \lim_{p \to \infty} Co^{(p)}(A)$ The crucial innovation is an algebraic operation "box-plus" ($\boxplus$), defined for scalars by: $$\lambda \boxplus \mu = \begin{cases} \lambda & |\lambda| > |\mu| \ \dfrac{1}{2}(\lambda + \mu) & |\lambda| = |\mu| \ \mu & |\lambda| < |\mu| \end{cases}$$ This binary operation is symmetric ($\lambda\boxplus\mu = \mu\boxplus\lambda$) and idempotent ($\lambda\boxplus\lambda = \lambda$) but generally non-associative: $$(\lambda\boxplus\mu)\boxplus\nu \neq \lambda\boxplus(\mu\boxplus\nu)$$ unless extra conditions hold.

To generalize to multiary combinations, an $n$-ary operation,

$\boxplus_{i \in I} x_i = \mathcal{D}_I(x)$

is constructed, where $\mathcal{D}_I(x)$ uses a determinant-like device that encodes the limiting procedure and allows expressing convex hulls of arbitrary size.

Elements of $Co^{(\infty)}(A)$ take the form: $$x = \boxplus_j \left( \boxplus_{\sigma \in S_{m+1}} \alpha_{\sigma,j} x^{(j)} \right)$$ where $\alpha_{\sigma,j}$ are coefficients derived from the ϕₚ-determinant, so that their sum (in the sense of $\boxplus$) over $j$ and $\sigma$ equals 1.

2. Idempotent, Non-Associative Algebraic Structure

The algebraic structure underpinning ℬ-convexity is defined by symmetric, idempotent, but non-associative operations. For binary, $\boxplus$ is as above; for arbitrary many points, the operation is extended using index-dependent determinant-like functions.

Distinct companion operations, $\boxminus$ and $\boxplus^+$, encode dominant sign and symmetry features:

• $$u \boxplus v = \dfrac{1}{2} (u \boxminus v + u \boxplus^+ v)$$

Such formalism allows encoding the "largest in modulus" operation—generalizing max-plus convexity to the whole vector space, not just the positive orthant. The determinant construction ensures unambiguous multi-point convex combinations despite lack of associativity.

3. Limiting Polytopes and Structural Properties

A "ℬ-polytope" is defined as the Kuratowski–Painlevé limit polytope associated to this operation. In the positive orthant, the result reduces to classical max-plus convexity: $$Co^{(\infty)}(A) = \left\{ \bigsqcup_{j} t_j x^{(j)} : t_j \in [0,1], \max_j t_j = 1 \right\}$$ where $\bigsqcup$ denotes the lattice maximum.

For general subsets of ℝⁿ, $Co^{(\infty)}(A)$ is described explicitly using the multiary $\boxplus$ and the determinant coefficients. Notably, for $m=2$ (two generators), the paper proves that the limit coincides with binary idempotent symmetric convexity studied in earlier work [b15], but for $m>2$ and higher $n$, the limit polytope need not satisfy idempotent symmetric convexity, illustrating strict generalization.

4. Separation Theorems in ℝⁿ

The paper establishes a separation property for ℬ-convex sets by approximating them through ℬ-polytopes. Specifically, given two finite sets $A$ and $E$ with $Co^{(\infty)}(A) \cap Co^{(\infty)}(E) = \emptyset$, one can construct, for sufficiently large $p$, hyperplanes separating $Co^{(p)}(A)$ and $Co^{(p)}(E)$. Taking limits, an idempotent linear form,

$$f(x) = \langle a, x \rangle_{\infty} = \bigsqcup_{i=1}^n (a_i x_i)$$

is identified. Its lower and upper semi-continuous regularizations $f^-$ and $f^+$ give the sets: $$Co^{(\infty)}(A) \subset \{ x : f^{-}(x) \leq c \}, \quad Co^{(\infty)}(E) \subset \{ x : f^{+}(x) \geq c \}$$ This algebraic separation method generalizes affine separation in the classical convexity setting.

5. External (Half-Space) Representation of ℬ-Polytopes

Extending classical polytope theory, the paper proves that ℬ-polytopes admit an external representation via finite families of halfspaces defined through lower (and upper) idempotent symmetric forms: $$Co^{(\infty)}(A) = \bigcap_{j=1}^\ell \{ x \in \mathbb{R}^n : f_j^-(x) \leq c_j \}$$ provided that the intersection is "topologically regular." In the case of $\mathbb{R}_+^n$, the polytope reduces to the classical max-times convex set.

6. Comparison with Previous Definitions and Limitations

Earlier definitions (such as [14]) described ℬ-convexity as a topological Kuratowski-type limit without explicit algebraic structure. Recent advances [9], [10], and the present framework provide direct algebraic operations capable of expressing multi-point convex hulls, particularly through nonassociative, determinant-indexed summations. Significantly, in higher dimensions or with more than two generating points, the limit polytopes do not generally satisfy the binary idempotent symmetric convexity, requiring the broader multiary formalism.

7. Conclusion: Structural and Practical Implications

The multiary algebraic form of ℬ-convexity establishes a rigorous framework for polytope generation, separability, and representability in Euclidean spaces using symmetric, idempotent, and nonassociative operations. The approach unifies classical convexity, max-plus tropical geometry, and new algebraic formulations, enabling separation theorems and external representations previously unattainable with topological definitions alone. This algebraic perspective sets the foundation for further paper of generalized convex structures, with anticipated impact on optimization, computational geometry, and tropical analysis.

References:

• (Briec, 18 Sep 2025) On the Multiary Algebraic Formulation of an Idempotent Symmetric Limit Convex Structure
• [b15], [bh], [9], [10], [14]: See cited works in (Briec, 18 Sep 2025) for historical context and prior developments.