If we have a metric space $ (X,d) $, a finite set of distinct points $ a_1, ..., a_n $, do the distance functions $ d(a_1,x), ..., d(a_n,x) $ have to be linearly independent?
That is, if $ c_1d(a_1,x) + ... + c_nd(a_n,x) = 0$ for all $ x $, then $ c_1 = ... = c_n = 0 $? This can easily be seen to be true for $ n = 2 $ and also for $ n = 3 $, but what's the answer in general?
If we substitude all the $ a_i $ for x, we get an interesting system of n equations, where we can treat $ c_i $ as the unknowns, and then the matrix of the coefficients $ d(a_i, a_j) $ is symmetric with 0s on the diagonal(apparently this is called a "hollow matrix"). All this looks quite interesting and promising, but I don't really know where to take it.