On regular vertices of the union of planar convex objects

Esther Ezra; János Pach; Micha Sharir

doi:10.1007/s00454-008-9118-2

On regular vertices of the union of planar convex objects

Esther Ezra, János Pach, Micha Sharir

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3 Scopus citations

Abstract

Let C be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in C intersect in at most s points for some constant s4. We show that the maximum number of regular vertices (intersection points of two boundaries that intersect twice) on the boundary of the union U of C is O*(n ^4/3), which improves earlier bounds due to Aronov et al. (Discrete Comput. Geom. 25, 203-220, 2001). The bound is nearly tight in the worst case. In this paper, a bound of the form O* (f(n)) means that the actual bound is C _ε f(n) n ^ε for any ε>0, where C _ε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0).

Original language	English
Pages (from-to)	216-231
Number of pages	16
Journal	Discrete and Computational Geometry
Volume	41
Issue number	2
DOIs	https://doi.org/10.1007/s00454-008-9118-2
State	Published - Mar 2009
Externally published	Yes

Bibliographical note

Funding Information:
Work by János Pach and Micha Sharir was supported by NSF Grant CCF-05-14079, and by a grant from the U.S.–Israeli Binational Science Foundation. Work by Esther Ezra and Micha Sharir was supported by grant 155/05 from the Israel Science Fund and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the first author has also been supported by an IBM Doctoral Fellowship. A preliminary version of this paper has been presented in Proc. 23nd Annu. ACM Sympos. Comput. Geom., 2007, pp. 220–226.

Funding

Work by János Pach and Micha Sharir was supported by NSF Grant CCF-05-14079, and by a grant from the U.S.–Israeli Binational Science Foundation. Work by Esther Ezra and Micha Sharir was supported by grant 155/05 from the Israel Science Fund and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the first author has also been supported by an IBM Doctoral Fellowship. A preliminary version of this paper has been presented in Proc. 23nd Annu. ACM Sympos. Comput. Geom., 2007, pp. 220–226.

Funders	Funder number
israel science fund
National Science Foundation	CCF-05-14079
International Business Machines Corporation
United States-Israel Binational Science Foundation	155/05
Tel Aviv University

Keywords

(1/r)-cuttings
Bi-clique decompositions
Geometric arrangements
Lower envelopes
Regular vertices
Union of planar regions

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10.1007/s00454-008-9118-2

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On regular vertices of the union of planar convex objects

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