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I have a problem similar to that of finding the geometric median. Suppose I have $n$ fixed points $x_1,x_2,\dots,x_n$; what is the set of points that have the same total distance $\ell$ to these fixed points?

Is there any theory relating to this problem? Is there any chance that these points have some relationship with the geometric median?

Parcly Taxel
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Ben
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  • I do not quite understand "the collection of points that have the same total distance to these points". Could you give a small example? Does this mean for example with two points and a stated total you get an ellipse/ellipsoid? – Henry Oct 04 '16 at 09:11
  • The smallest possible total distance will (by definition) correspond to the geometric median. – Henry Oct 04 '16 at 09:15
  • Hi Henry, thank you for your reply. I know about geometric median. However, I am not going to find the geometric median. Instead, I am going to find the all the points that the total distance between each point to the sample points is the same. – Ben Oct 04 '16 at 12:46

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The set of points that have the same sum-distance to $n$ given points is called an $n$-ellipse. An example is shown below.

Otherwise known as the Tschirnhaus egg curve

These curves have almost nothing to do with the geometric median, except that the median is what the curves degenerate to as the given sum-distance decreases. Maxwell studied them, and we know a few properties: they are polynomial plane curves of high degree and they can approximate certain convex closed curves.

I once investigated the area enclosed by such a curve, and the results were far from simple.

Parcly Taxel
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