Every finite metric space of strictly negative type admits a canonical signed measure (the microscopic weighting) that maximizes distance energy and equals the derivative of the magnitude function at zero
Stolarsky,Extremal problems of distance geometry related to energy integrals, Transactions of the American Mathematical Society193(1974), 1–31
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The microscopic weighting on a metric space
Every finite metric space of strictly negative type admits a canonical signed measure (the microscopic weighting) that maximizes distance energy and equals the derivative of the magnitude function at zero