On the cardinalities of a t-sets in a real Hilbert space

Abstract

Abstract A set X in a real Hilbert space H is called an a t-set if every three-element subset of X forms either an acute-angled triangle or a right-angled triangle. The maximal cardinality of an a t-set in an infinite-dimensional H is found. Furthermore, the number of right angles in the unit cube [ 0 , 1 ] n ${[0,1]^n}$ is calculated. As an application, a simple solution of a well-known problem is given, concerning the maximal cardinality of a strong a t-set in the Euclidean space ℝ n .