On The Ordered Sets In n-Dimensional Real inner Product Spaces

Abstract

Let X be a real inner product space of dimension ¸ 2. In [2], W. Benz proved the following theorem for x; y 2 X with x < y: "The Lorentz-Minkowski distance between x and y is zero (i.e., l (x; y) = 0) if and only if [x; y] is ordered". In this paper, we obtain necessary and su±cient conditions for Lorentz-Minkowski distances l(x; y) > 0; l (x; y) < 0 with the help of ordered sets in n-dimensional real inner product spaces.