# An alternative proof of Petty's theorem on equilateral sets

• Volume: 109, Issue: 2, page 165-175
• ISSN: 0066-2216

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## Abstract

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The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in a normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for any three points in a normed plane, forming an equilateral triangle of side p, there exists a fourth point, which is equidistant to the given points with distance not larger than p. We will also improve the example given by Petty and obtain the existence of a smooth and strictly convex norm in ℝⁿ for which there exists a maximal 4-element equilateral set. This shows that the theorem of Petty cannot be generalized to higher dimensions, even for smooth and strictly convex norms.

## How to cite

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Tomasz Kobos. "An alternative proof of Petty's theorem on equilateral sets." Annales Polonici Mathematici 109.2 (2013): 165-175. <http://eudml.org/doc/280981>.

@article{TomaszKobos2013,
abstract = {The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in a normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for any three points in a normed plane, forming an equilateral triangle of side p, there exists a fourth point, which is equidistant to the given points with distance not larger than p. We will also improve the example given by Petty and obtain the existence of a smooth and strictly convex norm in ℝⁿ for which there exists a maximal 4-element equilateral set. This shows that the theorem of Petty cannot be generalized to higher dimensions, even for smooth and strictly convex norms.},
author = {Tomasz Kobos},
journal = {Annales Polonici Mathematici},
keywords = {equilateral set; equilateral dimension; equidistant points; touching translates; norm; sphere; convex body},
language = {eng},
number = {2},
pages = {165-175},
title = {An alternative proof of Petty's theorem on equilateral sets},
url = {http://eudml.org/doc/280981},
volume = {109},
year = {2013},
}

TY - JOUR
AU - Tomasz Kobos
TI - An alternative proof of Petty's theorem on equilateral sets
JO - Annales Polonici Mathematici
PY - 2013
VL - 109
IS - 2
SP - 165
EP - 175
AB - The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in a normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for any three points in a normed plane, forming an equilateral triangle of side p, there exists a fourth point, which is equidistant to the given points with distance not larger than p. We will also improve the example given by Petty and obtain the existence of a smooth and strictly convex norm in ℝⁿ for which there exists a maximal 4-element equilateral set. This shows that the theorem of Petty cannot be generalized to higher dimensions, even for smooth and strictly convex norms.
LA - eng
KW - equilateral set; equilateral dimension; equidistant points; touching translates; norm; sphere; convex body
UR - http://eudml.org/doc/280981
ER -

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