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Ham Sandwich Is Equivalent to Borsuk-Ulam

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Ham Sandwich Is Equivalent to Borsuk-Ulam Ham Sandwich is Equivalent to Borsuk-Ulam Karthik C. S.∗1 and Arpan Saha†2 1 Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Israel [email protected] 2 Department of Mathematics, University of Hamburg, Germany [email protected] Abstract The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in Rn can always be simultaneously bisected by an (n − 1)-dimensional hyperplane. In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f : Sn → R, there exists a compact set A ⊆ Rn+1, such that for every x ∈ Sn we have f(x) = vol(A ∩ H+) − vol(A ∩ H−), where H is the oriented hyperplane containing the origin with ~x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics. Finally, using the above√ result we prove that there exist constants n0, ε0 > 0 such that for every n ≥ n0 and ε ≤ ε0/ 48n, any query algorithm to find an ε-bisecting (n − 1)-dimensional hyperplane of n compact sets in [−n4.51, n4.51]n, even with success probability 2−Ω(n), requires 2Ω(n) queries. 1998 ACM Subject Classification F.2.2 Computations on discrete structures, Geometrical prob- lems and computations Keywords and phrases Ham Sandwich theorem, Borsuk-Ulam theorem, Query Complexity, Hy- perspherical Harmonics Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.24 1 Introduction The Borsuk-Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point [6]. This result has countless applications in Mathematics [21]. In particular it implies the Brouwer’s Fixed Point Theorem [7, 16] which is the basis of several important results in Economics [5], for example Nash’s theorem [23]. Soon after the Borsuk-Ulam theorem was established, the Ham Sandwich theorem was proven using it [28, 29]. The Ham Sandwich theorem states that the volumes of any n compact sets in Rn can always be simultaneously bisected by an (n − 1)-dimensional hyperplane. However, as far as we know, there is no result in previous literature establishing the equivalence of the Borsuk-Ulam theorem and the Ham Sandwich theorem. ∗ This work was partially supported by Irit Dinur’s ISF-UGC 1399/14 grant. † This work was partially supported by the Research Training Group 1670. © Karthik C. S. and Arpan Saha; licensed under Creative Commons License CC-BY 33rd International Symposium on Computational Geometry (SoCG 2017). Editors: Boris Aronov and Matthew J. Katz; Article No. 24; pp. 24:1–24:15 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 24:2 Ham Sandwich is Equivalent to Borsuk-Ulam From a computational perspective, the computation of Brouwer fixed points has been studied extensively in various models of computations such as the Time/Computational complexity model [24, 12, 10, 9, 26, 27], the Query complexity model [18, 8, 11, 3, 27], and the Communication model [25, 4]. From these results one may obtain a reasonable understanding of the problem of computing equally valued antipodal points in a Borsuk-Ulam function by utilizing the constructive reduction from the Brouwer fixed point computation problem [30, 32]. However, computational aspects of the Ham Sandwich theorem have been poorly understood. In particular, no hardness result or non-trivial lower bounds in any model of computation are known in literature for the Ham Sandwich problem. In this paper, we prove that the Borsuk-Ulam theorem and the Ham Sandwich theorem are indeed equivalent! Moreover, we use this equivalence to prove a query complexity lower bound on the Ham Sandwich problem. 1.1 Our Results Our main result is a reduction from the Borsuk-Ulam theorem to the Ham Sandwich theorem. A key result in establishing the above reduction is that of establishing the equivalence between the two theorems for polynomial functions: n n I Proposition 1. For every polynomial f : S → R restricted to the hypersphere, there n+1 n exist n compact sets A1,...,An ⊆ R , such that for every x ∈ S and i ∈ [n], we have the following: + − fi(x) − fi(−x) = vol(Ai ∩ H ) − vol(Ai ∩ H ), th where fi(x) is the projection of f(x) to the i coordinate, and H is the oriented hyperplane containing the origin with ~x as the normal. After establishing the above result, we use the Stone-Weierstrass theorem to note that any continuous function can be arbitrarily well approximated by polynomial functions, and prove the Borsuk-Ulam theorem for all continuous functions. Next, we consider the Ham Sandwich problem in the query model: the input to the k kn problem is n compact sets A1,...,An ⊆ −n , n , for some constant k > 0, and each query + − is an oriented hyperplane H and the answer is vol(Ai ∩ H ) − vol(Ai ∩ H ), for all i ∈ [n]. The goal is to find a (n − 1)−dimensional hyperplane H such that each set is ε-bisected by + − H, i.e., for all i ∈ [n], we have |vol(Ai ∩ H ) − vol(Ai ∩ H )| ≤ ε. We show the following lower bound for the Ham Sandwich problem: √ I Theorem 2. There exist constants n0, ε0 > 0 such that for any n ≥ n0, ε ≤ ε0/ 48n, p = 2−Ω(n), and k ≥ 4.51 the following holds: any query algorithm to find an ε-bisecting n (n − 1)-dimensional hyperplane of n compact sets in −nk, nk , even with success probability p, requires 2Ω(n) queries. By assuming a notion of Lipschitz continuity, we can show that the number of queries needed to compute an ε−bisecting hyperplane is 2O(n log n) by querying translations of n hyperplanes over −nk, nk whose normals form an O(ε)-net over Sn. Thus, the above lower bound is tight up to logarithmic multiplicative factor in the exponent. Furthermore, we remark here that Theorem 2 is the first nontrivial lower bound for the Ham Sandwich problem in any model of computation. Karthik C. S. and A. Saha 24:3 1.2 Our Techniques and Proof Overview We provide below a proof-sketch of the reduction from the Borsuk-Ulam theorem to the Ham Sandwich theorem. The basic idea is to find for a given continuous odd function on Sn taking values in R, a compact measurable set in Rn+1, such that the given function is the difference of volumes of the set on the positive and negative side of an oriented hyperplane through the origin. This makes sense as the oriented hyperplanes through the origin are parametrised by Sn on which its positive unit normal takes values, so we actually get a continuous odd function on Sn. Then an oriented hyperplane bisects the set if and only if the given odd function vanishes at the point on Sn corresponding to the positive unit normal of the hyperplane. In particular, if we have have an odd continuous function from Sn to Rn, we can make the above argument for every component. Then an oriented hyperplane bisects the sets if and only if the given odd function taking values in Rn vanishes at the point on Sn corresponding to the positive unit normal of the hyperplane. A compact measurable set may be constructed by starting with a solid (n+1)-dimensional ball of unit radius centred at the origin and then radially scaling it by a continuous function on Sn taking values in R that is positive everywhere. Then the volume contained in a solid angle sector would be given by integrating an expression proportional to the (n + 1)-th power of the scaling function over the region on Sn corresponding to the solid angle sector. Thus the difference of volumes on either side of a hyperplane is related to the (n + 1)-th power of the scaling function by a linear integral transform. It turns out that this linear map becomes diagonal in the basis of hyperspherical harmonics and in order to invert this integral transform we work in this basis. Since the basis is infinite-dimensional, there may be issues with convergence. We tackle this by first constructing the inverse transform for functions that are restrictions of polynomial functions on Rn+1 to Sn (see Proposition 1), since in these cases, only finitely many elements in the basis suffice. This means that the reduction of Borsuk-Ulam theorem to the Ham Sandwich theorem holds for all functions that are restrictions of polynomial functions on Rn+1 to Sn. And then we use the Stone-Weierstrass theorem, which states that any continuous function on [−1, 1]n+1 may be uniformly approximated using polynomial functions, to extend the reduction to all continuous functions on Sn. In order to prove Theorem 2, we start from the randomized query complexity lower bound recently obtained by Rubinstein [27] (building on the works of Hirsch et al. [18] and Babichenko [3]) for the computation of approximate fixed points in a Brouwer function in the Euclidean norm. We then show that computing approximately equal-valued antipodal points in the query model is as hard as computing approximate fixed points in a Brouwer function by using Su’s constructive proof of the Brouwer fixed point theorem from the Borsuk-Ulam theorem [30].
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