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

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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	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

Keywords

Hyperplane arrangements
Zone theorem

All Science Journal Classification (ASJC) codes

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2014.12.001

Cite this

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abstract = "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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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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KW - Zone theorem

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DO - 10.1016/j.comgeo.2014.12.001

M3 - مقالة

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VL - 48

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EP - 341

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 4

ER -

On the zone of the boundary of a convex body

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

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