The diameter of a polygon is defined as the longest distance between any pair of vertices of the polygon.
Partition a unit equilateral triangle into seven regions so that all seven regions have the same polygon diameter.
How small can that diameter be made?
Inside the unit equilateral, draw a smaller equilateral with the same orientation and center. Draw the three altitudes for the larger triangle, but erase them inside the smaller triangle. This partitions the larger area into a triangle and 6 similar right parallelograms.
It is obvious to me that there is some size of interior triangle which will cause all 7 areas to have the same "polygon diameter". I did some arithmetic, and I came up a triangle size (and a polygon diameter) of Sqrt(13) - 3, which is approximately .60555.
Is this calculation correct? I am not sure, but it seemed correct?
Is this the smallest possible? I suspect so, but I have not tried to come up with other arrangements.