All-even sets in ℝ² and ℝ³

@misc{kalmanovich2026alleven,
  title = {{All-even sets in $\mathbbR^2$ and $\mathbbR^3$}},
  author = {Daniel Kalmanovich and Gabriel Nivasch},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38864},
  eprint = {2610.38864},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38864},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - All-even sets in ℝ^2 and ℝ^3
AU  - Daniel Kalmanovich
AU  - Gabriel Nivasch
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38864
UR  - https://arxiv.org/abs/2610.38864
AB  - Call a finite nonempty point set S⊂ℝ^d all-even if for every direction ℓ ∈ ℙ(ℝ^d), the number of hyperplanes perpendicular to ℓ 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∈ℝ^d line-all-even if for every direction ℓ ∈ ℙ(ℝ^d) the number of lines parallel to ℓ that intersect S is even. As Santos pointed out, every all-even set in ℝ^3 is line-all-even. We conjecture that there exist no line-all-even sets in ℝ^3 that affinely span 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.
ER  -