Square (0,0), (0,1), (1,1), (1,0)
In this example, the four fixed points are the vertices of a unit square. The constants inside toward the outside range from 2.84 to 4.0.
Given that the vertices are on a unit square then the minimum sum would be .
Repeating, the same sketch with a different graphing application, the values of K are from outside toward the center are 4.0, 3.75, 3.50, 3.25, 3.0, and 2.84
Trapezoid (-1,0), (0,1), (1,1), (2,0
The constant terms must be larger because the minimum sum of the distances is going to be greater than 4.
Here is the same problem using a different graphing application: The values of K from outside inward are 7.0, 6.75, 6.50, 6.25, 6.00, 5.75, 5.00, 4.75, and 4.5.
Further Explorations -
1. Try for points at the vertices of other quadrilaterals.
2. Try for points at the vertices of a non-convex quadrilateral.
3. Try for more that 4 points.