I recently came across a surprising theorem, due to Wolfgang Stadje, a special case of which states that:
Let $(X,d)$ be a compact connected metric space. Then there exists a unique real number $c$ such that for any positive integer $n$, and for any $x_1,\ldots,x_n\in X$, there exists $x\in X$ such that $$c=\frac{1}{n}\sum_{i=1}^n d(x,x_i)$$
The original proof of the general statement can be found here.
This proof is unaccessible to me. Since this proof was published in 1981, it is possible that simpler proofs have been found in subsequent years, especially for the special case mentioned above.
I haven't been able to find any other proof on the internet though.
Does anybody know a more accessible proof of the result mentioned above?