Abstract
Given a finite set of points \(S\subset {\mathbb {R}}^d\), a k-set of S is a subset \(A\subset S\) of size k which can be strictly separated from \(S\setminus A\) by a hyperplane. Similarly, a k-facet of a point set S in general position is a subset \(\varDelta \subset S\) of size d such that the hyperplane spanned by \(\varDelta \) has k points from S on one side. For a probability distribution P on \({\mathbb {R}}^d\), we study \(E_P(k,n)\), the expected number of k-facets of a sample of n random points from P. When P is a distribution on \({\mathbb {R}}^2\) such that the measure of every line is 0, we show that \(E_P(k,n)=O(n(k+1)^{1/4})\). Our argument is based on a technique by Bárány and Steiger. We study how it may be possible to improve this bound using the continuous version of the polynomial partitioning theorem. This motivates a question concerning the points of intersection of an algebraic curve and the k-edge graph of a set of points. We also study a variation on the k-set problem for the set system whose set of ranges consists of all translations of some strictly convex body in the plane. The motivation is to show that the technique by Bárány and Steiger is tight for a natural family of set systems. For any such set system, we determine bounds for the expected number of k-sets which are tight up to logarithmic factors.
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Notes
Recall we assume that hyperplanes have measure 0. In all our results we also restrict to Borel measures. The Borel assumption guarantees that every open halfspace is measurable which is necessary for (1.1) (and, in particular, the definition of \(G_P(t)\)) to make sense. In fact the assumption that P is Borel is the minimal assumption that guarantees that all halfspaces are measurable because the collection of open halfspaces generates the Borel sigma algebra on \({\mathbb {R}}^d\).
[13, Lem. 9.10] states that the number of convex chains is \(k-1\) and the number of concave chains is \(n-k+1\). The reason this doesn’t match the Lemma as we have stated it is that [13] defines a k-edge to be line segment connecting two points x, y such that there are k points in the closed halfspace below \({\text {aff}}(x,y)\) whereas we require that there are k points in the open halfspace below \({\text {aff}}(x,y)\). We choose open because it matches the standard definition of a k-facet.
We use the following version of the law of total probability: \({\mathbb {P}}(A\,|\,Y) ={\mathbb {E}}({\mathbb {P}}(A\,|\,X,Y)\,|\,Y)\). This follows from [7, Thm. 4.1.13 (ii)].
When \(k=({n-2})/{2}\) this counts each k-edge twice, so the constant in our bound can be improved in this case.
One could consider the same question for possibly singular curves, but, for our purposes, it suffices to consider non-singular curves.
The proof of Theorem 3.5 relies on the Stone–Tukey ham sandwich theorem [25] for \(L^1\) functions on \({\mathbb {R}}^d\). There is a version of the ham sandwich theorem which applies to more general distributions but it has a weaker conclusion and cannot be used to extend Theorem 3.5 to more general distributions as far as the authors know.
In order to apply VC’s uniform convergence theorem, we need to verify that the function \(\sup _{R\in {\mathcal {R}}}(\,{\cdot }\,)\) as defined in (4.1) is measurable, i.e., that it is a random variable. This can be verified by observing that \(\mathcal {R}\) is a permissible class of subsets of A. See [22, Appendix C] for the definition of permissible classes and a proof of the measurability of suprema in this context. One can see that the class \({\mathcal {R}}\) is permissible by indexing it by translation and verifying that the requirements for permissibility are met.
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Acknowledgements
We would like to thank the reviewers for their helpful suggestions. In particular, we are grateful to the anonymous reviewer who noticed that the bound in Theorem 1.3 could be significantly improved with a simple modification to the proof. This material is based upon work supported by the National Science Foundation under Grant Nos. CCF-2006994, CCF-1657939, CCF-1422830, and CCF-1934568.
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Leroux, B., Rademacher, L. Improved Bounds for the Expected Number of k-Sets. Discrete Comput Geom 70, 790–815 (2023). https://doi.org/10.1007/s00454-022-00469-7
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DOI: https://doi.org/10.1007/s00454-022-00469-7