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The problem is not well-defined for arbitrary point clouds. Clockwise around which point? How to handle collinear points? Does the path of sorted points need to have certain properties, e.g. no self-intersection? Here are some ideas, but they all have certain drawbacks: Pick a point $C$, such as the average of all points. Sort all points $P_i$ by the ...


The $L_1$ norm has a big flaw: try to apply $K$-clustering in a 2d space, with 2 centroids. When you have to say which points are near to which centroid there's a high grade of uncertainty, since the surface wiht the same distance from both centroids has infinite (lebesgue) measure. (on the contrary, euclidean distance has only a line, so a 0-measure set). ...

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