Minimizing variance of distances between points when mean distance is fixed - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:57:02Z http://mathoverflow.net/feeds/question/34896 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34896/minimizing-variance-of-distances-between-points-when-mean-distance-is-fixed Minimizing variance of distances between points when mean distance is fixed Robin Saunders 2010-08-08T06:31:03Z 2010-08-08T20:51:19Z <p>In R<sup>d</sup>, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d &isin; {1,2,3} although I'd be curious if there were any patterns for larger values of d. As for values of n, I'm interested both in specific solutions for small values, and general patterns in larger values.</p> http://mathoverflow.net/questions/34896/minimizing-variance-of-distances-between-points-when-mean-distance-is-fixed/34946#34946 Answer by Anton Petrunin for Minimizing variance of distances between points when mean distance is fixed Anton Petrunin 2010-08-08T20:51:19Z 2010-08-08T20:51:19Z <p>For \$d=1\$, it is minimized by arithmetic progression.</p>