On prescribing total orders and preorders to pairwise distances of points in Euclidean space

Víctor Hugo Almendra-Hernández; Leonardo Martínez-Sandoval

doi:10.1016/j.comgeo.2022.101898

On prescribing total orders and preorders to pairwise distances of points in Euclidean space

Víctor Hugo Almendra-Hernández, Leonardo Martínez-Sandoval

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

3 Scopus citations

Abstract

We show that any total preorder on a set of the form (X2) where X has n elements coincides with the order on pairwise distances of some point collection of size n in Rⁿ⁻¹. For total orders, a collection of n points in Rⁿ⁻² suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for total orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.

Original language	English
Article number	101898
Journal	Computational Geometry: Theory and Applications
Volume	107
DOIs	https://doi.org/10.1016/j.comgeo.2022.101898
State	Published - 1 Dec 2022
Externally published	Yes

Keywords

Convex sets
Euclidean distances
Total orders
Total preorders

ASJC Scopus subject areas

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

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10.1016/j.comgeo.2022.101898

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abstract = "We show that any total preorder on a set of the form (X2) where X has n elements coincides with the order on pairwise distances of some point collection of size n in Rn−1. For total orders, a collection of n points in Rn−2 suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for total orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.",

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AU - Almendra-Hernández, Víctor Hugo

AU - Martínez-Sandoval, Leonardo

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Y1 - 2022/12/1

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AB - We show that any total preorder on a set of the form (X2) where X has n elements coincides with the order on pairwise distances of some point collection of size n in Rn−1. For total orders, a collection of n points in Rn−2 suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for total orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.

KW - Convex sets

KW - Euclidean distances

KW - Total orders

KW - Total preorders

UR - http://www.scopus.com/inward/record.url?scp=85130341988&partnerID=8YFLogxK

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SN - 0925-7721

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JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 101898

ER -

On prescribing total orders and preorders to pairwise distances of points in Euclidean space

Abstract

Keywords

ASJC Scopus subject areas

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