It has been conjectured by H. S. Witsenhausen that the maximum M(d,n) of $\sum_{x,y \in X} \|x−y\|_2$ over all sets X consistingof n points in the d-dimensional Euclidean space with unit diameter is attained if and only if the points of X are distributed as evenly as possible among the vertices of a regular d-dimensional simplex of edge-length 1. In this paper the authors give a proof of this conjecture.
The sum of squared distances under a diameter constraint, in arbitrary dimension / Benassi, Carlo 6/8/1962; Malagoli, Federica. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - STAMPA. - 90:(2008), pp. 471-480. [10.1007/s00013-008-2509-z]
The sum of squared distances under a diameter constraint, in arbitrary dimension
BENASSI, Carlo 6/8/1962;MALAGOLI, Federica
2008
Abstract
It has been conjectured by H. S. Witsenhausen that the maximum M(d,n) of $\sum_{x,y \in X} \|x−y\|_2$ over all sets X consistingof n points in the d-dimensional Euclidean space with unit diameter is attained if and only if the points of X are distributed as evenly as possible among the vertices of a regular d-dimensional simplex of edge-length 1. In this paper the authors give a proof of this conjecture.
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