Helly-Type Theorems for Splitting Point Sets
Abstract
Let 0 < α≤ 1/2. We say that a finite point set P in ℝd is α-split by a hyperplane h if each of the closed half-spaces determined by h, contains at least α|P| of the points of P. We further say P is α-split by a k-dimensional flat τ if P is α-split by any hyperplane through τ. In the standard notation (which coincides with Tukey depth for k= 0), the k-flat τ has depth α with respect to P. We establish interesting Helly-type theorems for splitting families of finite point sets in ℝd. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of k-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.
Related
BibTeX
@misc{portal2026hellytype,
title = {{Helly-Type Theorems for Splitting Point Sets}},
author = {Lidor Portal and Natan Rubin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02180},
eprint = {2609.02180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02180},
}
RIS
TY - UNPB
TI - Helly-Type Theorems for Splitting Point Sets
AU - Lidor Portal
AU - Natan Rubin
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.02180
UR - https://arxiv.org/abs/2609.02180
AB - Let 0 < α≤ 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|P| of the points of P. We further say P is α-split by a k-dimensional flat τ if P is α-split by any hyperplane through τ. In the standard notation (which coincides with Tukey depth for k= 0), the k-flat τ has depth α with respect to P. We establish interesting Helly-type theorems for splitting families of finite point sets in ℝ^d. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of k-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.
ER -