Let $\displaystyle x$ and $\displaystyle y$ be distinct points in a Euclidean metric space. The set $\displaystyle \lbrace z \vert d(z,x) < d(z,y)\rbrace$ is convex. Does the same apply for any linear combination of the distances, i.e.: Is $\displaystyle \lbrace z \vert \lambda d(z,x) < (1 - \lambda) d(z,y)\rbrace$, where $\displaystyle 0 \leq \lambda \leq 1$, in any case convex?