Natan Rubin

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math.CO2026

Helly-Type Theorems for Splitting Point Sets

Lidor Portal, Natan Rubin

Let $0 < α \leq 1/2$ . We say that a finite point set $P$ in $R^{d}$ is $α$ -split by a hyperplane $h$ if each of the closed half-spaces determined by $h$ , contains at least $α…

math.CO2026

On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions

Natan Rubin

The 1913 Helly's theorem states that any family $K$ of $n \geq d + 1$ convex sets in $R^{d}$ can be pierced by a single point if and only if any $d + 1$ of $K$ 's e…

math.CO2024

Improved Bounds for Point Selections and Halving Hyperplanes in Higher Dimensions

Natan Rubin

Let $(P, E)$ be a $(d + 1)$ -uniform geometric hypergraph, where $P$ is an $n$ -point set in general position in $R^{d}$ and $E \subseteq (d + 1 P)$ is a collection of $ε{n\c…