What's the point in a polygon whose distances to all vertices sum up to be the least

In a triangle, it is the point such that when we draw the lines connecting to each vertex, the 3 adjacent angles are all 120 degrees. (Proof uses elementary geometry, try to spin things around and arrange the three attaching lines to a polygon arc and show it is the shortest when it's a straight line etc.)

When n is 4. The point is the intersection of the diagonal lines. Just connect any other point to all vertices and it is instant by triangular inequality that this is the smallest.

But what about pantogon, hexagon, etc etc?

Such a point obviously exist because the inside is compact and the sum of distance is continuous. But where is it?