How to find the best fit point inside a cluster?

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I have a cluster with many points.

Like this:

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Where I can visually identify a cluster of points and a noise point.

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Question: I would like to know how can I find the best fit point via an algorithm/math formulas. Not the centroid of the cluster, but the best fit point... probably the closest point to the cluster centroid without taking into account the noise point(s).

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The PAM algorithm (for partition around medoïd) is an algorithm close to the usual k-means which allows to works with data points as cluster centers.