For integers d≥ 2 and n≥ d+2, let F_1,…,F_n⊆R^d be sets, each of which is the union of exactly two disjoint, nonempty, closed convex sets. If, for every I⊆1,…,n with | I|∈1,2,3,d+1, the intersection bigcap_i∈ IF_i consists of exactly two nonempty closed convex components, must the total intersection bigcap_i=1^nF_i also consist of exactly two nonempty closed convex components?

Gil Kalai’s Helly-Type Question for Two-Component Convex Sets

For integers $d\ge 2$ and $n\ge d+2$ , let $F_1,\ldots,F_n\subseteq\mathbb{R}^d$ be sets, each of which is the union of exactly two disjoint, nonempty, closed convex sets. If, for every $I\subseteq\{1,\ldots,n\}$ with $\lvert I\rvert\in\{1,2,3,d+1\}$ , the intersection $\bigcap_{i\in I}F_i$ consists of exactly two nonempty closed convex components, must the total intersection $\bigcap_{i=1}^nF_i$ also consist of exactly two nonempty closed convex components?

Progress summary

Solved

A 2026 preprint claims to settle the question positively in dimensions two and higher, while exhibiting a one-dimensional counterexample.

Gil Kalai posed this question in the $2020$ Discrete Geometry meeting’s open-problems collection: does the required two-piece structure for sufficiently small intersections force the whole intersection to be nonempty? The claimed theorem gives a stronger conclusion, preserving exactly two pieces globally.

2026 claimed proof

Menara’s preprint claims that for $d\geq 2$ and $n\geq d+2$ , it suffices to check intersections of sizes $1$ , $2$ , $3$ , and $d+1$ ; the full intersection then has exactly two nonempty closed convex components. The proof labels components over $\mathbb{F}_2$ , uses triple compatibility, and applies Helly’s theorem. It also gives four subsets of $\mathbb{R}$ showing that $d=1$ fails. No independent verification, objection, or retraction was found, so the claim remains unverified.

Current status (as of August 2026): The positive result is claimed in an arXiv preprint for $d\geq 2$ , with $d=1$ disproved by an explicit example; formal acceptance and independent verification remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

A Helly-Type Theorem for two-component convex sets — arXiv — Menara, Giuliamaria

convex-geometry discrete-geometry

1 day ago