TWO-DISTANCE PRESERVING FUNCTIONS FROM EUCLIDEAN SPACE

Károly Bezdek; Robert Connelly

doi:10.1023/a:1004859411072

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 39: Issue 1-3

TWO-DISTANCE PRESERVING FUNCTIONS FROM EUCLIDEAN SPACE

Authors:

Károly Bezdek

Károly Bezdek
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and

Robert Connelly

Robert Connelly
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Pages:: 185–200

Online Publication Date:: 28 Dec 2004

Publication Date:: 01 Dec 2000

Article Category:: Research Article

DOI:: https://doi.org/10.1023/a:1004859411072

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Abstract

Let 0 < c < s be fixed real numbers such that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${c \mathord{\left/ {\vphantom {c s}} \right. \kern-\nulldelimiterspace} s} \leqslant {{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ \end{document}

, and let f : E² → E^d for d ≥ 2 be a function such that for every p, q ∈ E² if p − q = c, then f(p) − f(q) ≤ c, and if p − q = s, then f(p) − f(q) ≥ s. Then f is a congruence. This result depends on and expands a result of Rdo et. al. [9], where a similar result holds, but for

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 3}} \right. \kern-\nulldelimiterspace} 3}$$ \end{document}

replacing

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ \end{document}

. We also present a further extensions where E² is replaced by Eⁿ for n > 2 and where the range of c/s is enlarged.

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)