TITLE: Function as sum of distances over a connected, compact metric space
QUESTION [6 upvotes]: If $X$ is a connected, compact metric space with distance function $d : X^2 \rightarrow \mathbb{R}^+$, is it true that there exists a positive real number $a$, dependent on $X$ and $d$, such that for any $n$ and for any $x_1, x_2, \cdots, x_n \in X$, there exists $y$ such that $$\frac{1}{n}\left(d(x_1,y) + d(x_2,y) + \cdots + d(x_n,y)\right) = a?$$
Motivation: trivial when $X$ is the boundary of a circle, a tricky contest problem when $X$ is the boundary of a square (these examples all use Euclidean distance)

REPLY [5 votes]: It's a classic theorem of O. Gross from 1964. The number $a$ is also unique for a given space $(X,d)$.
There is an exposition at https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Cleary-Morris-Yost260-275.pdf 
The original paper is:
O. Gross, "The rendezvous value of a metric space", Advances in Game Theory, Ann. of Math. Studies no. 52, Princeton, 1964, 49-53