Six of us were having dinner in a restaurant in on Friday evening. We had travelled from different parts of London and had travelled different decisions. I started thinking about where we could have met to minimise the total distance we travelled.
In other words, given a set of friends and their starting points and assuming we measure straight-line distances - ignore roads and hills - there must be at least one point that requires the least total distance to be travelled in order for everybody to meet. Furthermore, there may be more than one such point. As an example, consider just two friends: any point on a straight line between them would be a meeting point that satisfied my requirement that the total distance travelled is miminised.
My question is: given an arbitrary number of friends and their starting points, what can we say about the set of points representing minimum total distances travelled? In particular, will its points necessarily be connected to each other or could they be dispersed?