The 3-diametral number of Euclidean d-space is 2d+2

A configuration of n points in ℝd, repetitions allowed, is called 3-diametral if among any three of its points two are at distance equal to the diameter of the configuration. Bezdek and Lángi proved that the largest possible n is 2d+2 for d ≤ 3, noted that Polyanskii's spectral method gives 2d+4 in general, and asked whether 2d+2 holds in every dimension. We prove that it does, and that the vertices of a regular simplex, each taken twice, form the only extremal configuration. Consequently a 3-diametral set of distinct points in ℝd has at most 2d+1 points. The proof encodes a configuration of diameter 1 by the matrix W = (1 − |xi − xj|2), which is the Gram matrix of n unit vectors in a Lorentzian space of signature (d+1, 1), and combines the trace identity tr W3 = 3 tr W2 − 2n with the Perron–Frobenius inequality for the nonnegative matrix W − I. The equality case rests on two lemmas about pairs of regular simplices of edge length 1 lying on a common sphere or in a common flat whose union has diameter 1. For distinct points we construct 8 points in ℝ4 and d+1+2⌊(d+1)/3⌋ points in ℝd, so that Polyanskii's conjectured bound ⌊3(d+1)/2⌋ fails for every d ≥ 4 with d ≠ 7.