An alternative proof of Petty's theorem on equilateral sets

Volume 109 / 2013

Tomasz Kobos Annales Polonici Mathematici 109 (2013), 165-175 MSC: 46B85, 46B20, 52C17, 52A15, 52A20. DOI: 10.4064/ap109-2-5


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.


  • Tomasz KobosFaculty of Mathematics and Computer Science
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland

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