Average point in a Banach Space

Question

In $\mathbb{R}^n$, we get the average of several vectors $x_1,...,x_n$ by adjusting $\bar{x}$ to minimize the sum of squares: $$ \sum_{i=1}^n d(\bar{x},x_i)^2. $$ Do we have similar result for general metric space? Is there any articles about it?

Answer 1

The result is true in any inner product space. In fact any finite dimensional subspace of an inner product space can be identified with $\mathbb R^{n}$ for some n, so no new proof is required. In a general metric space average of elements does not make sense and there is no general formula for the element that minimizes the sum. Even in $\mathbb R^{2}$ if you change the metric to say $d((a,b),(x,y))=|x-a|+|y-b|$ the result is not true.