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Given a finite set $S = \lbrace x: x \in R^n \rbrace $ of points, where the $i$-th point is associated with a weight $w_i \in [0,1]$ and all these points are spread across the surface of a n-sphere $S^n$. I perform a stereographic projection to $R^n$ and notice that there is also a distance metric (as shown below) that I can make use of: Metric on n-sphere in terms of stereographic projection coordinates

My problem is that I need a weighted-metric, whereby the weights associated with each point are also taken into account. So, for example, given any arbitrary point $p \in S^n$ and $d$ is the metric (unweighted) on the sphere, the nearest point $p^*$ to $p$ would minimize $d(p,p^*)$ and maximize $w_{p^*}$. In other words, the metric must favour points that are closer to $p$ on the sphere, and whose weights are the largest.

I am wondering if such a min-max objective function can be achieved in a metric. I was thinking of something simple like : $\alpha \cdot d(p,x) + (1-\alpha)\cdot w_x$ to start with. Are there better ways to do this on a Riemannian manifold like the sphere?

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