Calculating the central point with minimal average distance to other points

Question

I work at an office with colleagues coming from all over the country. Our office is quite centrally located, but some colleagues have to travel quite a lot further than others. I often wondered how I could calculate a central point which minimizes the average traveling distance for each employee (traveling as the crow flies). So if we're ever going to relocate, that would be the ideal spot; we could all save on time and fuel.

Look at this image:

Here we can see that the employees live in different places. With 2 employees, it's rather simple. Let's pick John and Pete. The point halfway between John and Pete would be the perfect spot for them. But how about when we include a 3rd person or an n'th? I'm kind of lost there.

Bonus points for explaining it in a way an average but not expert mathematician understands. :)

Answer 1

@Davio

You might consider the centroid which is easier to calculate and defined similarly to the geometric median as minimizing the sum of the squares of the distances to each point and can be found by a simple formula. https://en.wikipedia.org/wiki/Geometric_median

Remember though this is for a plane. It does appear that using a program such as Geogebra I can minimize the distances in an empirical fashion and you might want to investigate that.

Answer 2

The average distance is the sum of distances divided by the number of colleagues. Since the latter is fixed, you can as well ask for the point which minimizes the sum. Which, by the way, indicates that for two employees the situation would be not as simple as you make it to be, since any point on the connecting line will satisfy the requirement.

Taking the terms “point minimize distance sum” to Wikipedia, you can find that such a point is called the geometric median of your set of employee locations. If you continue reading, you will find that computing it might be tricky, but there has been work on the subject.