Acute Sets In Euclidean Spaces

Abstract

A finite set in is called an acute set if any angle determined by three points of is acute. We examine the maximal cardinality of a -dimensional acute set. The exact value of is known only for . For each we improve on the best known lower bound for . We present different approaches. On one hand, we give a probabilistic proof that . (This improves a random construction given by Erdo˝s and Füredi.) On the other hand, we give an almost exponential constructive example which outdoes the random construction in low dimension . Both approaches use the small dimensional examples that we found partly by hand (, 5) and partly by computer . We also investigate the following variant of the above problem: what is the maximal size of a -dimensional cubic acute set (that is, an acute set contained in the vertex set of a -dimensional hypercube)? We give an almost exponential constructive lower bound, and we improve on the best known upper bound.