Around the Petty theorem on equilateral sets

TL;DR

This paper offers an alternative proof of Petty's theorem on equilateral sets in normed spaces, extends the understanding of equilateral set extensions, and demonstrates limitations in higher dimensions with smooth, strictly convex norms.

Contribution

Provides a new proof of Petty's theorem, extends results to the plane, and shows the non-generalizability to higher dimensions with smooth, strictly convex norms.

Findings

Every 3-point equilateral set in a normed plane can be extended to 4 points.

Constructs a smooth, strictly convex norm with a maximal 4-element equilateral set in higher dimensions.

Shows Petty's theorem cannot be extended to higher dimensions even with smooth, strictly convex norms.

Abstract

The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in the 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 N\'emeth about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for every 3 points in the normed plane, forming an equilateral set of the common distance $p$, there exists a fourth point, which is equidistant to the given points with the 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 $\mathbb{R}^n$, which contain 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.