An alternative proof of Petty's theorem on equilateral sets

Tomasz Kobos¹

¹ Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 109 (2013) no. 2, pp. 165-175

Résumé

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 $\mathbb {R}^n$ 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.

Détail

DOI : 10.4064/ap109-2-5

Keywords: main paper provide alternative proof following theorem petty normed space dimension least three every element equilateral set extended element equilateral set approach based result kramer meth about inscribing simplex convex body prove theorem petty shall establish three points normed plane forming equilateral triangle side there exists fourth point which equidistant given points distance larger improve example given petty obtain existence smooth strictly convex norm mathbb which there exists maximal element equilateral set shows theorem petty cannot generalized higher dimensions even smooth strictly convex norms