All-even point sets in planes and three-dimensional spaces studied
All-even sets in $\mathbb{R}^2$ and $\mathbb{R}^3$
Computational Geometry
Summary
The paper studies special groups of points called all-even sets where every direction sees an even number of planes or lines hitting the points. The authors explore these sets in two and three dimensions and propose that no such sets exist in three-dimensional space that span it fully. They relate this idea to a property called height-rigidity, which they investigate through examples and mathematical equivalences. This work clarifies the structure of these unusual point sets and contributes to understanding geometric configurations.
What this means in practice
- •For computational geometers: Use the characterization of all-even sets to analyze geometric properties and constraints in point arrangement problems.
- •For mathematical software developers: Incorporate conditions like height-rigidity into tools for geometric modeling and verifying spatial point set properties.
A theory result. No direct application yet.
Authors
Daniel Kalmanovich, Gabriel Nivasch
Abstract
Call a finite nonempty point set $S\subset\mathbb{R}^d$ all-even if for every direction $\ell \in \mathbb{P}(\mathbb{R}^d)$, the number of hyperplanes perpendicular to $\ell$ that intersect $S$ is even. In 2026, Francisco Santos presented a question of Juan Benigno Seoane-Sepúlveda on the existence of all-even sets in dimensions $3$ and higher, together with examples of such sets in dimension $2$. Call a set $S\in\mathbb{R}^d$ line-all-even if for every direction $\ell \in \mathbb{P}(\mathbb{R}^d)$ the number of lines parallel to $\ell$ that intersect $S$ is even. As Santos pointed out, every all-even set in $\mathbb{R}^3$ is line-all-even. We conjecture that there exist no line-all-even sets in $\mathbb{R}^3$ that affinely span $\mathbb{R}^3$ We show that our conjecture is equivalent to the statement that every planar all-even set satisfies a certain property, which we call height-rigidity. This property has a natural interpretation in linear algebra terms. Furthermore, we present a hierarchy of properties of planar point sets that imply height-rigidity. We construct several all-even sets in the plane, including some infinite families, and show that all the sets we constructed are height-rigid. In addition, we prove that every all-even set in the plane that can be covered by six or fewer parallel lines is height-rigid.