Fractional Helly theorem for Cartesian products of convex sets
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2021
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| _version_ | 1866917584583000064 |
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| author | Chakraborti, Debsoumya Kim, Jaehoon Kim, Jinha Kim, Minki Liu, Hong |
| author_facet | Chakraborti, Debsoumya Kim, Jaehoon Kim, Jinha Kim, Minki Liu, Hong |
| contents | Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $α\in (1-\frac{1}{t^d},1]$ and a finite family $\mathcal{F}$ of Cartesian products of convex sets $\prod_{i\in[t]}A_i$ in $\mathbb{R}^{td}$ with $A_i\subset \mathbb{R}^d$ if at least $α$-fraction of the $(d+1)$-tuples in $\mathcal{F}$ are intersecting then at least $(1-(t^d(1-α))^{1/(d+1)})$-fraction of sets in $\mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound $α>1-\frac{1}{t^d}$ and the fraction $(1-(t^d(1-α))^{1/(d+1)})$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $\mathbb{R}^d$ does not have $(p,d+1)$-condition for sublinear $p$. Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional $(p,d+1)$-condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_09962 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Fractional Helly theorem for Cartesian products of convex sets Chakraborti, Debsoumya Kim, Jaehoon Kim, Jinha Kim, Minki Liu, Hong Combinatorics 05E45, 52A35 Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $α\in (1-\frac{1}{t^d},1]$ and a finite family $\mathcal{F}$ of Cartesian products of convex sets $\prod_{i\in[t]}A_i$ in $\mathbb{R}^{td}$ with $A_i\subset \mathbb{R}^d$ if at least $α$-fraction of the $(d+1)$-tuples in $\mathcal{F}$ are intersecting then at least $(1-(t^d(1-α))^{1/(d+1)})$-fraction of sets in $\mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound $α>1-\frac{1}{t^d}$ and the fraction $(1-(t^d(1-α))^{1/(d+1)})$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $\mathbb{R}^d$ does not have $(p,d+1)$-condition for sublinear $p$. Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional $(p,d+1)$-condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable. |
| title | Fractional Helly theorem for Cartesian products of convex sets |
| topic | Combinatorics 05E45, 52A35 |
| url | https://arxiv.org/abs/2108.09962 |