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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap109-2-5,
author = {Tomasz Kobos},
title = {An alternative proof of Petty's theorem on equilateral sets},
journal = {Annales Polonici Mathematici},
volume = {107},
year = {2013},
pages = {165-175},
zbl = {1295.46016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap109-2-5}
}
Tomasz Kobos. An alternative proof of Petty's theorem on equilateral sets. Annales Polonici Mathematici, Tome 107 (2013) pp. 165-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap109-2-5/