A weighted geometric regularity from order restricted statistical inference

Abstract

In Euclideank-space, the cone of vectors x = (x 1,x 2,...,x k ) satisfyingx 1 ≤x 2 ≤ ... ≤x k and\(\sum\nolimits_{j = 1}^k {x_j } = 0\) is generated by the vectorsv j = (j −k, ...,j −k,j, ...,j) havingj −k’s in its firstj coordinates andj’s for the remainingk −j coordinates, for 1 ≤j <k. In this equal weights case, the average angle between v i and v j over all pairs (i, j) with 1 ≤i <j <k is known to be 60°. This paper generalizes the problem by considering arbitrary weights with permutations.