# What is the center of minimum distance of a region?

Suppose we have a compact plane region $$R$$ (not necessarily convex or connected). I am working in a problem which involves the point $$p$$ in $$R$$ that is, in average, the closest to every other point. That is, the point in $$R$$ which minimizes $$\int_R d(p,x) dx.$$ I have been looking for references but I can't find this point anywhere (it is certainly not the centroid / baricenter, as the centroid doesn't even need to be a point in $$R$$).

Do you know if this point has a name or has been studied previously?

• Continuous version of "medoid"? en.wikipedia.org/wiki/Medoid – Jukka Kohonen May 6 at 9:39
• @JukkaKohonen Thank you, that is exactly what I was looking for. If you post it as an answer I will accept it. – Gutiérrez May 6 at 10:39
• This question came up previously for triangles in particular -- and even for that simple shape, the calculations were so involved that it still has not gone into the Encyclopedia of Triangle Centers. See mathoverflow.net/questions/240186/… – Matt F. May 7 at 16:57

There are two closely related notions: geometric median and medoid. Each minimizes the sum (or integral) of distances from the points in some set $$A$$. The difference is simply that medoid is additionally required to be in $$A$$, but a geometric median can (and in some cases will) be outside $$A$$. Finding either point is usually done by iterative algorithms.
I believe these are more often encountered in discrete settings ($$A$$ is a finite collection of points; often the case in descriptive statistics and also facility location) but perhaps with these terms you find something of the continuous versions too. One of the references from Wikipedia would, at least judging from its title, seem relevant: Fekete et al (2005), On the Continuous Fermat-Weber Problem, https://pubsonline.informs.org/doi/abs/10.1287/opre.1040.0137