Line segment without a midpoint
Task number: 3164
Find an example of a metric space, in which the midpoint of a line segment does not exist. In other words find a metric space \((X, \rho)\) and points \(x, y \in X\) such that, there does not exit an \(s\) which satisfies \(\rho(x, s) = \rho(y, s) = \frac12 \rho(x, y)\).
Resolution
Consider a metric space with two points \(X = \{1, 2\}\), with the metric \(\rho(1, 1) = \rho(2, 2) = 0\) and \(\rho(1, 2) = \rho(2, 1) = 1\). Choose \(x = 1\) a \(y=2\). The point \(s\) cannot exist, because in this space the length \(\frac12\) does not exist.
