Non-Separable Homothetic Triangles, Part I: Constructions and Lower Bounds
Abstract
A finite family of planar convex bodies is called a non-separable family if no line disjoint from its union has at least one member in each open half-plane. In this paper, we prove that there exist finite non-separable families of positive homothetic triangles with covering factors strictly exceeding the sharp three-member bound μ= 2/3 + 2/3√3. This resolves negatively a question posed by K. Bezdek and Z. Lángi, who originally established this three-member bound after proving that the classic factor 1 covering theorem by A. W. Goodman and R. E. Goodman for disks fails for arbitrary positive homothets. Besides giving explicit algebraic examples with four, five, and six members having factors of approximately 1.0533161, 1.0551900, and 1.0572061 respectively, we provide a common cyclic recurrence that yields a 303-member family with the exact factor 250000000/235141779. Finally, we derive a continuous model suggested by increasingly fine recurrences, yielding a numerical candidate of 1.0633083; its attainability and optimality remain open.
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BibTeX
@misc{lian2026nonseparable,
title = {{Non-Separable Homothetic Triangles, Part I: Constructions and Lower Bounds}},
author = {Yanlu Lian and Fei Xue and Qihui Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06558},
eprint = {2609.06558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06558},
}
RIS
TY - UNPB
TI - Non-Separable Homothetic Triangles, Part I: Constructions and Lower Bounds
AU - Yanlu Lian
AU - Fei Xue
AU - Qihui Yuan
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06558
UR - https://arxiv.org/abs/2609.06558
AB - A finite family of planar convex bodies is called a non-separable family if no line disjoint from its union has at least one member in each open half-plane. In this paper, we prove that there exist finite non-separable families of positive homothetic triangles with covering factors strictly exceeding the sharp three-member bound μ= 2/3 + 2/3√3. This resolves negatively a question posed by K. Bezdek and Z. Lángi, who originally established this three-member bound after proving that the classic factor 1 covering theorem by A. W. Goodman and R. E. Goodman for disks fails for arbitrary positive homothets. Besides giving explicit algebraic examples with four, five, and six members having factors of approximately 1.0533161, 1.0551900, and 1.0572061 respectively, we provide a common cyclic recurrence that yields a 303-member family with the exact factor 250000000/235141779. Finally, we derive a continuous model suggested by increasingly fine recurrences, yielding a numerical candidate of 1.0633083; its attainability and optimality remain open.
ER -