On the zone of the boundary of a convex body

Orit E. Raz

doi:10.1016/j.comgeo.2014.12.001

On the zone of the boundary of a convex body

Orit E. Raz^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We consider an arrangement A of n hyperplanes in ℝ^d and the zone Z in A of the boundary of an arbitrary convex set in Rd in such an arrangement. We show that, whereas the combinatorial complexity of Z is known only to be O(n^d-1 log n) [3], the outer part of the zone has complexity O(n^d-1) (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for d=2).

Original language	American English
Pages (from-to)	333-341
Number of pages	9
Journal	Computational Geometry: Theory and Applications
Volume	48
Issue number	4
DOIs	https://doi.org/10.1016/j.comgeo.2014.12.001
State	Published - May 2015
Externally published	Yes

Bibliographical note

Funding Information:
Work on this paper has been supported by Grant 338/09 from the Israel Science Fund and by the Hermann Minkowski MINERVA Center for Geometry at Tel Aviv University.

Publisher Copyright:
© 2014, Elsevier B.V. All rights reserved.

Keywords

Hyperplane arrangements
Zone theorem

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10.1016/j.comgeo.2014.12.001

Cite this

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N2 - We consider an arrangement A of n hyperplanes in ℝd and the zone Z in A of the boundary of an arbitrary convex set in Rd in such an arrangement. We show that, whereas the combinatorial complexity of Z is known only to be O(nd-1 log n) [3], the outer part of the zone has complexity O(nd-1) (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for d=2).

AB - We consider an arrangement A of n hyperplanes in ℝd and the zone Z in A of the boundary of an arbitrary convex set in Rd in such an arrangement. We show that, whereas the combinatorial complexity of Z is known only to be O(nd-1 log n) [3], the outer part of the zone has complexity O(nd-1) (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for d=2).

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JF - Computational Geometry: Theory and Applications

IS - 4

ER -

On the zone of the boundary of a convex body

Abstract

Bibliographical note

Keywords

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