What is $\min\limits_{{v_i\in R^m,\ \|v_i\|=1,}\atop{i\in\{1,2,\cdots,n\}}}\max\limits_{i,j}v_i^Tv_j$ where $\|\cdot\|$ denotes the Euclidean length? Basically, I would like to distribute a given finite number of unit points "uniformly" on a unit sphere.

A computationally easier formulation is $$\max_{{v_i\in R^m,\ \|v_i\|=1,}\atop{i\in\{1,2,\cdots,n\}}}\sum_{i,j}\|v_i-v_j\|^2.$$

Are these the two formulations equivalent? My guess is that they are not. Can anyone give a proof either way?


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