An alternative proof of Petty's theorem on equilateral sets
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.
- 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
- 46B20: Geometry and structure of normed linear spaces
- 52A15: Convex sets in 3 dimensions (including convex surfaces)
- 52A20: Convex sets in n dimensions (including convex hypersurfaces)
- 52C17: Packing and covering in n dimensions