K-Levels of Concave Surfaces

Katoh,; Tokuyama,

doi:10.1007/s00454-001-0086-z

Cited by 4 publications

(1 citation statement)

References 19 publications

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“…For instance, up to a multiplicative constant, the bound on the number of k-sets in the plane is equivalent to a bound on the number of vertices of the k-level of a simple arrangement of n lines. The k-levels of concave surfaces were considered in [KaT02].…”

Section: Problemmentioning

confidence: 99%

Problems on Points in General Position

Research Problems in Discrete Geometry

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In this section, just as in the rest of Chapter 8, all point sets are assumed to be in general position, that is, in the plane, they do not contain three points on a line, or, in d-dimensional space, d + 1 points in a hyperplane. These point sets can be described by their order types, as defined in Section 8.1.Perhaps the most famous problem concerning point sets in general position is the "Erdős-Szekeres convex polygon problem," also known as Esther Klein's problem or the Happy End problem* (see [MoS00], [BáK01], for recent surveys). What is the smallest number f ES (r) such that any set of f ES (r) points in general position in the plane contains r points that form the vertex set of a convex r-gon? Instead of saying that a point set contains the "vertex set of a convex r-gon," in the sequel we simply say that it contains a "convex r-gon."In their first joint paper, Erdős and Szekeres [ErS35] established the existence of this number by reducing the result to a Ramsey-type statement and rediscovering Ramsey's theorem for its proof. In a much later paper [ErS60], the same authors obtained the lower bound f ES (r) ≥ 2 r−2 + 1. The only known exact values of this function are f ES (3) = 3, f ES (4) = 5, and f ES (5) = 9 [Bo74], [KaKS70]. It is conjectured that the lower bound is sharp for all r. Conjecture 1 (Erdős-Szekeres [ErS35]) Any set of 2 r−2 + 1 points in general position in the plane contains a convex r-gon.

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Section: Problemmentioning

confidence: 99%

Problems on Points in General Position

Research Problems in Discrete Geometry

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On Levels in Arrangements of Surfaces in Three Dimensions

Chan

2012

Discrete Comput Geom

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A favorite open problem in combinatorial geometry is to determine the worst-case complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n "pseudo-planes" or "pseudo-spherical patches" (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most O(n 2.997 ) vertices at any given level.

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Low-dimensional linear programming with violations

Chan

The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.

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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most violations: finding a point inside all but at most of Ò given halfspaces. We give a simple algorithm in 2-d that runs in Ç´´Ò · ¾ µ Ð Ó Òµ expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matoušek (1994) and is probably nearoptimal for all Ò ¾. A (theoretical) extension of our algorithm in 3-d runs in near Ç´Ò · ½½ Ò ½ µ expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the´ µ-level, previously used in proving combinatorial -level bounds.Applications in the plane include improved algorithms for finding a line that misclassifies the fewest among a set of bichromatic points, and finding the smallest circle enclosing all but points. We also discuss related problems of finding local minima in levels. LP with violations: the problem and background.We can define the problem of LP with at most violations in dimensions as follows:Given a set À of Ò halfspaces in IR , a linear objective function , and an integer ¼ Ò ¾, minimize over the region Á ´Àµ Õ ¾ IR Õ lies outside halfspaces of À or report that Á ´Àµ .Since our interest is in geometric applications, we confine our discussion to small constant values of . (The problem for arbitrary dimensions is NP-complete; see for example the references at [22] under the name "Maximum Hyperplane Consistency.") Note that Á ¼´À µ, the intersection of all halfspaces, is the feasible region of the original LP problem. Following Matoušek [39], we call the special case of the problem in which Á ¼´À µ the feasible case. In the feasible case (where by

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K-Levels of Concave Surfaces

Cited by 4 publications

References 19 publications

Problems on Points in General Position

Problems on Points in General Position

On Levels in Arrangements of Surfaces in Three Dimensions

Low-dimensional linear programming with violations

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