Convexity in topological affine planes

Raghavan Dhandapani; Jacob E. Goodman; Andreas Holmsen; Richard Pollack; Shakhar Smorodinsky

doi:10.1007/s00454-007-1336-5

Convexity in topological affine planes

Raghavan Dhandapani
, Jacob E. Goodman
, Andreas Holmsen
, Richard Pollack
, Shakhar Smorodinsky

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

7 Scopus citations

Abstract

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.

Original language	English
Pages (from-to)	243-257
Number of pages	15
Journal	Discrete and Computational Geometry
Volume	38
Issue number	2
DOIs	https://doi.org/10.1007/s00454-007-1336-5
State	Published - 1 Jan 2007
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-007-1336-5

Cite this

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AU - Holmsen, Andreas

AU - Pollack, Richard

AU - Smorodinsky, Shakhar

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Y1 - 2007/1/1

N2 - We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.

AB - We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.

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Convexity in topological affine planes

Abstract

ASJC Scopus subject areas

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