Generalizing ham sandwich cuts to equitable subdivisions

Abstract

We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points and qm blue points in the plane in general position, there exists an equitable svbdivision of the plane into q disjoint convex polygons, each of which contains n red points and m blue points. For q = 2 this Theorem is equivalent to the Ham Sandwich Theorem. We also present an efficient algorithm for constructing an equitable subdivision.

Near equipartitions of colored point sets

“…We may therefore assume that H i contains strictly less than i(b + 1) points or strictly more than i(b + 1) points. The following observation is well-known (see for instance [3,Lemma 3]) and can be shown by a simple continuity argument. 1), then there exists a halfplane H ′′ i satisfying |H ′′ i ∩ A| = ia and |H ′′ i ∩ B| = i(b + 1).…”

“…The planar case of Theorem 5 has the following generalization, conjectured by Kaneko and Kano [7], and proven independently by Bespamyatnikh et al [3], Ito et al [6] and Sakai [17].…”

“…which violates condition (1). [3] assumes that a 1 /b 1 = a 2 /b 2 = a 3 /b 3 , but Kaneko et al [9] observed that the proof can be easily modified so that the assumption can be omitted. See Figure 3.…”

Plane Bichromatic Trees of Low Degree

“…In Section 2 we prove Conjecture 1 for the case when |R| = k|B|, with k 1. The proof-which is very simple-is based on a result of Bespamyatnikh et al [3] and a recent result of Hoffmann et al [7]. The core of our contribution is in Section 3, where we partially prove Conjecture 2: If |R| = k|B|, with k 2, and R ∪ B is in general position, then there exists a plane bichromatic tree on R ∪ B whose maximum degree is k + 1.…”

Convex Equipartitions of Colored Point Sets

“…Theorem 1 answers the case when k = m = d. The case m ≥ k = d = 2 was settled by Aichholzer et al [1] and by Kano, Suzuki and Uno [8]. Further developments on the planar case were made independently by Bespamyatnikh, Kirkpatrick and Snoeyink [3], Ito, Uehara and Yokoyama [6] as well as Sakai [11], who confirmed the conjecture for two colors (m = d = 2) when the sizes of the color classes are divisible by n. Holmsen, Kynčl and Valculescu resolved the conjecture for the remaining cases in the plane, as well as for the case when k − 1 = m = d ≥ 2, the latter by giving a particular discretization of the ham-sandwich theorem [5]. Their method is similar to the one used previously by Kano and Kynčl [7] to establish the case m − 1 = d = k, who for the proof developed a generalization of the ham-sandwich theorem for d + 1 measures in R d , which they called the hamburger theorem.…”