Acute Sets In Euclidean Spaces

A finite set H in Rd is called an acute set if any angle determined by three points of H is acute. We examine the maximal cardinality α(d) of a d-dimensional acute set. The exact value of α(d) is known only for d ≤ 3. For each d ≥ 4 we improve on the best known lower bound for α(d). We present different approaches. On one hand, we give a probabilistic proof that α(d) > c · 1.2d. (This improves a random construction given by Erdős and Füredi.) On the other hand, we give an almost exponential constructive example which outdoes the random construction in low dimension (d ≤ 250). Both approaches use the small dimensional examples that we found partly by hand (d = 4, 5), partly by computer (6 ≤ d ≤ 10). We also investigate the following variant of the above problem: what is the maximal size κ(d) of a d-dimensional cubic acute set (that is, an acute set contained in the vertex set of a d-dimensional hypercube). We give an almost exponential constructive lower bound, and we improve on the best known upper bound. 2010 Mathematics Subject Classification: 05D40, 51M16.