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As for the sum of squares (m=2), denoting the vectors $\overline{OB_i}=b_i$ we get $\sum_{i<j} y_{ij}^2=\sum_{i<j} (b_i-b_j)^2=n\sum_i b_i^2-(\sum b_i)^2=n^2R^2-(\sum b_i)^2$ that is maximal if and only if $\sum b_i=0$ --- so, in particular, for a regular polygon.