All-even sets in $\mathbb{R}^2$ and $\mathbb{R}^3$

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.