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Bounding a global red-blue proportion using local conditions Ma´rton Naszo´di ∗1, Leonardo Mart´ınez-Sandoval †2, and Shakhar Smorodinsky‡3 1 Department of Geometry, Lorand Eötvös University, Budapest, Hungary 1EPFL, Lausanne, Switzerland 2Department of Computer Science, Ben-Gurion University of the Negev, Be’er-Sheva Israel. 7 3Department of Mathematics, Ben-Gurion University of the Negev, Be’er-Sheva Israel. 1 0 2 February 24, 2017 b e F 2 Abstract thateachclientisassociatedwithsomedisk 2 centered at the client’s location and hav- ] We study the following local-to-global phe- ing radius representing how far in the plane G nomenon: Let B and R be two finite sets his device can communicate. Suppose also, C of (blue and red) points in the Euclidean that some communication protocol requires . s plane R2. Suppose that in each “neighbor- that in each of the clients disks, the num- c [ hood” of a red point, the number of blue ber of antennas is at least some fixed pro- 2 points is at least as large as the number of portion λ > 0 of the number of clients in v red points. We show that in this case the the disk. Our question is: does such a lo- 0 0 total number of blue points is at least one cal requirement imply a global lower bound 2 fifth of the total number of red points. We on the number of antennas in terms of the 2 0 also show that this bound is optimal and number of clients? In this paper we answer . we generalize the result to arbitrary dimen- thisquestionandprovideexactbounds. Let 1 0 sion and arbitrary norm using results from us formulate the problem more precisely. 7 Minkowski arrangements. Let B and R = {p ,...,p } be two finite 1 1 n : setsinR2. LetD = {D ,...,D }beasetof v 1 n i Euclidean disks centered at the red points, X 1 Introduction i.e., the center of D is p . Let {ρ ,...,ρ } r i i 1 n a be the radii of the disks in D. Consider the following scenario in wireless Theorem 1.1. Assume that for each i we networks. Suppose we have n clients and have |D ∩B| ≥ |D ∩R|. Then |B| ≥ n. m antennas where both are represented as i i 5 Furthermore, the multiplicative constant 1 points in the plane (see Figure 1). Each 5 cannot be improved. client has a wireless device that can communicate with the antennas. Assume also Such a local-to-global ratio phenomenon was shown to be useful in a more combina- ∗Email address: [email protected]. †Email address: [email protected]. torial setting. Pach et. al. [PRT15], solved ‡Email address: [email protected]. a conjecture by Richter and Thomassen 1 Minkowski arrangements see, e.g., [FL94]. We need the following auxiliary Lemma. Lemma 1.2. Let K be an origin-symmetric convex body in Rd. Let R = {p ,...,p } be 1 n a set of points in Rd and let D = {K = 1 p + ρ K,...,K = p + ρ K} be a fam- 1 1 n n n ily of homothets of K. Then there exists a subfamily D(cid:48) ⊂ D that covers R and forms a strict Minkowski arrangement. Moreover, D(cid:48) can be found using a greedy algorithm. As a corollary, we will obtain the following theorem. Figure 1: In each device range (each disk) Theorem 1.3. Let K be an origin- there are at least as many antennas (black symmetric convex body in Rd. Let R = dots)asdevices(whitedots),sothehypoth- {p ,...,p } be a set of points in Rd and esis holds for λ = 1. 1 n let D = {K = p + ρ K,...,K = p + 1 1 1 n n ρ K} be a family of homothets of K where n [RT95] on the number of total “crossings” ρ ,...,ρ > 0. Let B be another set of 1 n that a family of pairwise intersecting curves points in Rd, and assume that, for some in the plane in general position can have. λ > 0, we have Lemma 1 from their paper is a first step in the proof and it consists of a local-to-global |B ∩K | i ≥ λ, (1) phenomenon as described above. |R∩K | i We will obtain Theorem 1.1 from a more general result. In order to state it, we in- for all i ∈ [n]. Then |B| ≥ λ . |R| 3d troduce some terminology. Let K be an origin-symmetric convex In Theorem 1.1 the convex body K is a body in Rd, that is, the unit ball of a norm. Euclidean unit disk in the plane. Another case of special interest is when the convex Astrict Minkowski arrangement isafam- body K is a unit cube and thus it induces ily D = {K = p + ρ K,...,K = p + 1 1 1 n n ρ K} of homothets of K, where p ∈ Rd the (cid:96)∞ norm. In this situation we get a n i sharper and optimal inequality. and ρ > 0, such that no member of the i family contains the center of another mem- Theorem 1.4. If K is the unit cube in ber. An intersecting family is a family of Rd, then the conclusion in Theorem 1.3 can sets that all share at least one element. be strengthened to |B| ≥ λ . Furthermore, We denote the maximum cardinality of |R| 2d the multiplicative constant 1 cannot be im- an intersecting strict Minkowski arrange- 2d proved. ment of homothets of K by M(K). It is known that M(K) exists for every K In the results above, the points p play i and M(K) ≤ 3d (see, e.g., Lemma 21 of the role of the centers of the sets of the [NPS16]). On the other hand (somewhat Minkowski arrangement. One might ask if surprisingly), there is an origin-symmetric thisrestrictionisessential. Asafinalresult, convex body K in Rd such that M(K) = (cid:16)√ (cid:17) we give a general construction to show that d Ω 7 , [Tal98, NPS16]. For more on it is. 2 follows: Add K to D(cid:48). Among all red 1 points that are not already covered by D(cid:48) pick a point p whose corresponding homo- j thet K has maximum homothety ratio ρ . j j Add K to D(cid:48) and repeat until all red points j are covered by D(cid:48). Note that the homothets in D(cid:48) are not necessarily disjoint. Clearly, R ⊂ (cid:83)D(cid:48). Now we show that no member of D(cid:48) contains the center of another. Suppose to the contrary that K i contains the center of K . If i < j, then j ρ ≥ ρ so K was chosen first, a contradic- i j i tion to the fact that p was chosen among j the points not covered by previous homo- Figure 2: The centers of the disks are la- thets. If i > j, then K also contains the beled in decreasing order of corresponding j center of K , and we get a similar contra- radii. The shaded disks cover the white i diction. points and no shaded disk contains the cen- This finishes the proof of Lemma 1.2. ter of another. Theorem 1.5. Let K be any convex body in Proof of Theorem 1.3. By Lemma 1.2, the plane and ε,λ any positive real numbers. there exists a subfamily D(cid:48) ⊂ D that There exist sets of points R = {p ,...,p } covers R and form a strict Minkowski 1 n and B in the plane such that |B| < εn and arrangement. Namely, (cid:83)D(cid:48) covers R, and that for each i there is a translate K of K no point of B is contained in more than i that contains p for which |B∩K | ≥ λ|R∩ M(K) members of D(cid:48). In particular, it i i K |. follows that i In particular, even if each red point is (cid:88) (cid:88) |B ∩K| M(k) |R| ≤ |R∩K| ≤ ≤ |B| contained in a unit disk with many blue λ λ K∈D(cid:48) K∈D(cid:48) points, theglobalbluetoredratiocanbeas small as desired. This is a possibly counter- so |B| λ λ intuitive fact in view of Theorem 1.1. ≥ ≥ . |R| M(K) 3d This completes the proof. 2 Proofs Lemma 2.1. Let K be the Euclidean unit Proof of Lemma 1.2. We construct a sub- disk centered at the origin. Then M(K) = family D(cid:48) of D with the property that no 5. member of D(cid:48) contains the center of any member of D(cid:48), and (cid:83)D(cid:48) covers the red Proof of Lemma 2.1. Five unit disks cen- points, R. Assume without loss of gener- tered in the vertices of a unit-radius regular ality that the labels of the points in R are pentagon show that M(K) ≥ 5. See Figure sorted in non-increasing order of the homo- 3a. thety ratio, that is, ρ ≥ ··· ≥ ρ . See To prove the other direction, suppose 1 n Figure 2 for an example. that there is a point b in the plane that is We construct D(cid:48) in a greedy manner as contained in 6 Euclidean disks in a strict 3 centers u and v. By applying a rotation we may assume that it is the region of vectors with non-negative entries. We may also assume δ := (cid:107)u(cid:107) ≥ (cid:107)v(cid:107) . ∞ ∞ Since the d-cube centered at u contains the origin, its radius must be at least δ. We claim that this cube contains v. Indeed, eachoftheentriesofuandv areintheinter- val [0,δ]. So each of the entries of u−v are in [−δ,δ]. Then (cid:107)u−v(cid:107) ≤ δ as claimed. ∞ This contradiction finishes the proof. Theorem 1.1 clearly follows from combin- ing the proof of Theorem 1.3 (with λ = 1) and Lemma 2.1. The result is sharp be- Figure3: OptimalMinkowskiarrangements cause we have equality when R is the set of in the plane for a) Euclidean disks, b) axis- vertices of a regular pentagon with center p parallel squares. and B = {p}. Similarly, Theorem 1.4 and its optimality follow from Lemma 2.2. Minkowski arrangement. Then, by the pi- geonhole principle, there are two centers of Remark 2.3. Lemma 2.1 can be general- those disks, say p and q such that the an- ized to arbitrary dimension. This implies gle (cid:94)(pbq) is at most 60◦. Assume without that Theorem 1.1 can be generalized to ar- loss of generality that pb ≥ qb. It is easily bitrary dimension almost verbatim. verified e.g., by the law of cosines, that the distance pq is less than pb. Hence, the disk centered at p contains q, a contradiction. Proof of Theorem 1.5. Let K be any con- This completes the proof. vex body in the plane. We construct sets R and B as follows. Let (cid:96) be a tangent line of Lemma 2.2. Let K be the unit cube of Rd K which intersects K at exactly one point centered at the origin. Then M(K) = 2d. t. Let I be a non-degenerate closed line segment contained in K and parallel to (cid:96). Let Proof of Lemma 2.2. Let d be a positive in- J be the (closed) segment that is the locus teger and e ,e ,...,e the canonical base of 1 2 n Rd. Consider all the cubes of radius 1 cen- of the point t as K varies through all its translations in direction d that contain I. tered at each point of the form ±e ±e ± 1 2 ...±e . ThisfamilyshowsthatM(K) ≥ 2d. See Figure 4. d See Figure 3b for an example on the plane. We construct R by taking any n points Now we show the other direction. Con- from J and we construct B by taking any siderthe2d closedregionsofRd boundedby mpointsfromI. ForanypointinR thereis the hyperplanes x = 0 i = 1,2,...,d and a translation of K that contains exactly one i supposeonthecontrarythatwehaveanex- point of R and m points of B, which makes ample with 2d +1 cubes or more that con- the local B to R ratio equal to m. But tain the origin. By the pidgeon-hole princi- globallywecanmaketheratio m arbitrarily n ple there is a region with at least two cube small. 4 References [FL94] Z. Fu¨redi and P. A. Loeb, On the best constant for the Besicovitch cover- ing theorem,Proc.Amer.Math.Soc.121 (1994), no. 4, 1063–1073. MR1249875 (95b:28003) [NPS16] M. Naszódi, J. Pach, and K. Swanepoel, Arrangements of homothets of a convex body,arXiv:1608.04639[math](2016). arXiv: 1608.04639. [PRT15] J. Pach, N. Rubin, and G. Tardos, Be- yond the Richter-Thomassen conjecture, arXiv:1504.08250 [math] (2015). arXiv: 1504.08250. Figure 4: Construction of example without [RT95] R. B. Richter and C. Thomassen, Inter- local-to-global phenomenon. sectionsofcurvesystemsandthecrossing number of C ×C , Discrete & Computa- 5 5 tional Geometry 13 (1995), no. 2, 149– 159. Acknowledgements [Tal98] I. Talata, Exponential lower bound for the translative kissing numbers of d- dimensional convex bodies,DiscreteCom- put. Geom. 19 (1998), no. 3, Special Is- M.Naszódiacknowledgesthesupportofthe sue, 447–455. Dedicated to the memory Jańos Bolyai Research Scholarship of the of Paul Erd˝os. MR98k:52046 Hungarian Academy of Sciences, and the National Research, Development, and Inno- vation Office, NKFIH Grant PD-104744, as well as the support of the Swiss National Science Foundation grants 200020-144531 and 200020-162884. L. Martinez-Sandoval’s research was partially carried out during the author’s visit at EPFL. The project leading to this ap- plication has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 re- searchandinnovationprogrammegrantNo. 678765 and from the Israel Science Founda- tion grant No. 1452/15. S. Smorodinsky’s research was partially supported by Grant 635/16 from the Israel Science Foundation. A part of this research was carried out during the author’s visit at EPFL, supported by Swiss National Sci- ence Foundation grants 200020-162884 and 200021-165977. 5