Empty Simplices in Euclidean Space

Abstract

Let P - {p 1 ,p 2 ,. . . ,p n } be an independent point-set in ℝ d (i.e., there are no d + 1 on a hyperplane). A simplex determined by d + 1 different points of P is called empty if it contains no point of P in its interior. Denote the number of empty simplices in P by f d ( P ). Katchalski and Meir pointed out that . Here a random construction P n is given with , where K( d ) is a constant depending only on d . Several related questions are investigated.