Geometric median of two disjoint sets of points lies on line between their respective medians

Question

I was working on a problem about geometric medians and I had an idea for a divide and conquer solution, but it would only work if a set of points, when split into two disjoint sets, and those sets respective geometric medians calculated and a line passed between, it holds that the geometric median of the overall set of points lies on that line. When graphing it and visualizing it, seems possible, and when I used existing algorithms and experimented, it seems to be very close but not quite on the same line. I am not sure if this is due to error in calculation of the geometric median or not.

I have some example Python code demonstrating how close the numbers are here: Geometric Median comparison

Any expertise in this matter would be appreciated.

Answer 1

If $A$ and $B$ are two set of points, the geometric median of $A \cup B$ does not necessarily lie on the line passing between the geometric mean of A and the geometric mean of $B$.

In the following example the points of $A$ are red and the one in $B$ is blue. The geometric median of $A$ is the red square. The green line connects the geometric means of $A$ and $B$. The black square is the geometric mean of $A \cup B$, i.e., the Fermat point of the triangle whose vertices are in $A \cup B$.