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An alternative proof of Petty's theorem on equilateral sets

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Kobos T.

Annales Polonici Mathematici

2013

tom 109

nr 2

165-175

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Kobos T.

1 2013

Wyniki wyszukiwania

An alternative proof of Petty's theorem on equilateral sets