Let and be distinct points in a Euclidean metric space. The set is convex. Does the same apply for any linear combination of the distances, i.e.: Is , where , in any case convex?
It's clear to me that for the simple case that , the set must be convex. To generate the said set, you just need to draw a line through the two points, half it, and draw a perpendicular line through the half. The halfplane on the side of is then the said set, which is convex. But I did not manage to generalise the proof.