The sum of squared distances under a diameter constraint, in arbitrary dimension

Abstract

Let X = {x1, ..., xn} be a set of n points in the d-dimensional Euclidean space E d , with unit diameter. In this work we give the complete proof of a conjecture by Witsenhausen, stating that the maximum M (d, n) of n i,j=1 xi − xj 2 in dimension d is attained if and only if the points are distributed as evenly as possible among the vertices of a regular d-dimensional simplex of edge-length 1. Mathematics Subject Classification (2000). Primary, 52A40.

Convergence in a Multidimensional Randomized Keynesian Beauty Contest

“…We remark that (1/n)G n is the squared radius of gyration of x 1 , . , x n ) := G n−1 (X n ) = G n−1 (x (1) , . .…”

“…, X N−1 instead of X (1) 13) since N ≥ 3. Here and elsewhere, 1(A) denotes the indicator random variable of the event A.…”

“…Denote the corresponding increasing order statistics U [1] ≤ · · · ≤ U [n] , and define the induced spacings by S n,i : .…”

“…Randomized Keynesian beauty contest 79 So, for example, the first three moments of D 2 (X 3 (0)) are 1 8 , 1 40 , and 1 160 , while the first three moments of μ 2 (X 3 (0)) are 1 2 , 51 160 , and 73 320 . Finally, E[μ 2 (X 3 (0))(D 2 (X 3 (0))) 2 ] = 1 80 .…”

Convergence in a Multidimensional Randomized Keynesian Beauty Contest

“…We write μ n (X n ) := n −1 n i=1 x i for the barycentre of X n , and • for the Euclidean norm on R d . Let ord(X n ) = (x (1) , x (2) , . .…”