Let us assume that one has four points with masses on the unit circle and the masses sum up to 1.
What is an optimal configuration (with optimal weights) if we want to maximize the sum of ${\it weighted}$ distances from the barycentre?
With formulas: let $P_i$-s be the points of weights $w_i \geq 0$ on the unit circle such that $\sum_{i=1}^4 w_i = 1.$ What is the maximum of $$ \sum_{i=1}^4 w_i \mbox{dist}(P_i, B), $$ where $B$ is the barycentre of $P_i$-s?
A rather natural idea is to choose a square with masses 1/4 at each vertex. But this is really an optimal solution? How this can be formalized?
Remark: A similar question was asked here math.stackexchange.com/q/1538308. Relying on google search engine, the precise answer there was given by L. Fejes Tóth ("On the sum of distances determined by a pointset", Acta Math. Acad. Sci. Hungar., 7:397–401, 1956).