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Let $X\subseteq\mathbf{R}^2$ be open, bounded, non-empty and define $c_p$ as the set of points minimizing $\int_{x\in X} \|x-c\|^p$, so e.g. $c_2$ contains only the centroid of $X$.

I have the intuition that if there is a unique largest circle inscribed into $X$, then $\lim_{p\to 0}|c_p|$ converges to $1$ and the point inside converges to the center of that circle.

Conversly, I imagine that $\lim_{p\to\infty}|c_p|$ converges to $1$ and the point inside converges to the center of the smallest circle containing $X$ (which is always unique).

Are there any results leading in this direction? What can be said about the the topology of $\cup_{0<p<\infty}c_p$ in general?

EDIT

OK, I just found this question and I think my first conjecture for $p\to 0$ is just wrong. It seems to me now that for the unit ball with a very small hole at, say, $1/2$ away from the center, this limit, if it exists, would be closer to the center of the ball than to the center of the smallest inscribed circle, which would be around $3/8$ away from the center if I'm not mistaken.

fweth
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  • What is the meaning of the absolute value symbols in $|c_p|$ ? If it the number of elements, how can you be sure that $c_p$ is finite ? – Jean Marie Feb 09 '23 at 13:20
  • Yes, the number of elements. It doesn't need to be finite. – fweth Feb 09 '23 at 13:23
  • IMHO, you must be much more restrictive about $X$ ; being bounded and open is way not enough : if it has "holes" for example, nothing can be said ... You should begin for example by a triangle, and more generaly by the interior of a polygon. – Jean Marie Feb 09 '23 at 13:36
  • Why are you sure it wouldn't work with holes? I'm more confident about the $p\to\infty$ case, I think I could draft a proof for this one (and arbitrary $X$), but for $p\to 0$ it is more complicated, I agree. I still think that e.g. a circle with a hole away from the center would work. – fweth Feb 09 '23 at 14:09

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