Hyperspaces of max-plus convex subsets of powers of the real line

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^\tau$ of the real line. We show that the corresponding spaces are AR’s if and only if $\tau\le\omega_1$.

💡 Research Summary

The paper investigates the topology of hyperspaces consisting of max‑plus convex subsets of the Tychonoff powers ℝ^τ of the real line. A max‑plus convex set is defined with respect to the operations ⊕ (maximum) and ⊗ (addition): a subset C ⊂ ℝ^τ is max‑plus convex if it is closed under the formation of max‑linear combinations a⊗x ⊕ b⊗y for any scalars a,b∈ℝ and points x,y∈C. This notion generalizes ordinary convexity to the tropical (max‑plus) algebraic setting and is natural in optimization and scheduling problems where “taking the maximum” replaces linear averaging.

The authors consider the collection 𝔐(ℝ^τ) of all non‑empty closed max‑plus convex subsets of ℝ^τ, equipped with the Vietoris topology. This topology coincides with the usual hyperspace topology on the space 𝔎(ℝ^τ) of all non‑empty closed subsets, but the additional convexity constraint yields a proper subspace. The first technical step is to show that 𝔐(ℝ^τ) is a complete metric space and that the inclusion i:𝔐(ℝ^τ)→𝔎(ℝ^τ) is continuous. Consequently, many standard tools from hyperspace theory (e.g., inverse limits, selection theorems) become available.

The central theorem states that 𝔐(ℝ^τ) is an absolute retract (AR) if and only if the cardinal τ does not exceed the first uncountable cardinal ω₁. The proof splits into two directions:

1. Sufficiency (τ ≤ ω₁).
   For each point x∈ℝ^τ the authors construct its max‑plus convex hull conv₊{x}. By normalising each coordinate into the unit interval, they obtain a continuous embedding of 𝔐(ℝ^τ) into the Hilbert cube Q =