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A Note on Smaller Fractional Helly Numbers

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Abstract

Let \({\mathcal {F}}\) be a family of geometric objects in \({\mathbb {R}}^d\) such that the complexity (number of faces of all dimensions on the boundary) of the union of any m of them is \(o(m^k)\). We show that \({\mathcal {F}}\), as well as \(\{F \cap P \mid F \in {\mathcal {F}}\}\) for any given set \(P \in {\mathbb {R}}^d\), have fractional Helly number at most k. This improves the known bounds for fractional Helly numbers of many families.

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Acknowledgments

The investigation of the relation between the union complexity and the fractional Helly number arose while working with Eyal Ackerman on a different problem. This study was supported by ISF grant (Grant No. 1357/12).

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  1. Faculty of Mathematics, The Technion, 32000, Haifa, Israel

    Rom Pinchasi

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  1. Rom Pinchasi
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Correspondence to Rom Pinchasi.

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Editor in Charge: János Pach

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Pinchasi, R. A Note on Smaller Fractional Helly Numbers. Discrete Comput Geom 54, 663–668 (2015). https://doi.org/10.1007/s00454-015-9712-z

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  • DOI: https://doi.org/10.1007/s00454-015-9712-z

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