A system of particles distributed on the surface of a ball, what is the "center of mass" of them on the surface?

The great circle distance between two points $\ c\ $ and $\ p_i\ $ on the surface of a ball is proportional to the angle subtended by the points at centre of the ball—that is, to $$ \arccos\left(\frac{\langle c,p_i\rangle}{\|c\|\,\|p_i\|}\right)\ . $$ To simplify matters, choose your unit of distance to be the radius of the ball, so that $\ \|p_i\|=1\ $ for all $\ i\ $, and the condition $\ c\in\Omega\ $ is equivalent to $\ \|c\|^2=1\ $. Your optimisation problem then reduces to \begin{align} &\min_{c\in\mathbb{R}^3}\sum_{i=1}^n m_i\arccos\big(\langle c,p_i\rangle\big)^2\\ &\text{subject to: }\ \|c\|^2=1\ . \end{align} The Lagrange condition for the minimum is $$ {\Large\sum_{i=1}^n}\frac{m_i\arccos\big(\langle c,p_i\rangle\big) p_i}{\sqrt{1-\langle c,p_i\rangle^2}}-\lambda c=0\ . $$ From this and the condition $\ \|c\|^2=1\ $, we get $$ \lambda^2={\Large\sum_{i=1}^n}{\Large\sum_{j=1}^n}\frac{m_im_j\arccos\big(\langle c,p_i\rangle\big)\arccos\big(\langle c,p_j\rangle\big)\langle p_i,p_j\rangle}{\sqrt{1-\langle c,p_i\rangle^2}\sqrt{1-\langle c,p_j\rangle^2}}\ . $$ Since we're looking for the minimum of the objective, the optimal $\ c\ $ must be in the opposite direction to its gradient, so \begin{align} \lambda&=-\sqrt{{\Large\sum_{i=1}^n}{\Large\sum_{j=1}^n}\frac{m_im_j\arccos\big(\langle c,p_i\rangle\big)\arccos\big(\langle c,p_j\rangle\big)\langle p_i,p_j\rangle}{\sqrt{1-\langle c,p_i\rangle^2}\sqrt{1-\langle c,p_j\rangle^2}}}\ . \end{align} If we take $\ w_i=\langle c,p_i\rangle\ $ as a set of $\ n\ $ unknowns in the above equations, they must satisfy the following $\ n\ $ simultaneous non-linear equations $$ w_j=\lambda(w)^{-1}{\Large\sum_{i=1}^n}\frac{m_i\arccos\big(w_i\big) \langle p_i,p_j\rangle}{\sqrt{1-w_i^2}}\ . $$ I doubt if there's any simple expression for the solution. I expect the best you'd be able to do is solve them numerically for any given values of the $\ m_i\ $ and $\ p_i\ $. Alternatively, it's probably going to be simpler to solve the optimisation problem directly with a gradient projection method.