Large regular simplices contained in a hypercube

We prove that the n-dimensional unit hypercube contains an n-dimensional regular simplex of edge length c √ n, where c > 0 is a constant independent of n. Let l∆n be the n-dimensional regular simplex of edge length l, and let lQn be the n-dimensional hypercube of edge length l. For simplicity, we omit l if l= 1, e.g., Qn denotes the unit hypercube. We are interested in the maximum edge length of a regular n-dimensional simplex contained in Qn. Theorem. For every ε0 > 0 there is an N0 such that for every n > N0 one has ( 1− ε0 2 √ n ) ∆n ⊂ Qn. On the other hand, if l∆n ⊂ Qn, then l ≤ √ (n+1)/2, which follows by comparing the circumscribed balls of l∆n and Qn. (Recall that the circumradius of ∆n is √ n/(2n+2).) This upper bound is reached iff there exists an Hadamard matrix of order n+1. Schoenberg [3] pointed out that this “readily established fact” went back to Coxeter, see also §4 of [1]. Our lower bound given by the theorem is approximately 1/ √ 2 of the upper bound. Proof of Theorem. For a matrix (or a vector) A=(ai j), let us define its norm by ∥A∥ := maxi j |ai j|. Let Jn be the n×n all one matrix. Lemma 1. Let A = (ai j) be an n×n real orthogonal matrix, and let c > 0 be a constant. If ∥A∥ ≤ 1 c √ n (1) Date: November 18, 2008. 2000 Mathematics Subject Classification. Primary: 52C07 Secondary: 05B20.