An alternative proof of Petty's theorem on equilateral sets
Volume 109 / 2013
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 $\mathbb {R}^n$ 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.
