Two pairs of centers for convex regions

Recording some experimental work done over the last week or so. A discussion page on this is at: https://math.stackexchange.com/questions/4482893/two-possible-triangle-centers

For every point P in a planar convex region C, we consider (1) the shortest and longest chords of C that pass thru P and (2) the shortest and longest distances from P to the boundary of C.

Define: 'Chord ratio(P)' = ratio between lengths of longest and shortest chords thru P. 'Chord difference(P)' = difference between lengths of longest and shortest chords thru P.

It is observed that when C is a triangle, both above quantities have a minimum value at the same interior point except for very flat triangles (this exceptional case was observed rather late so I was tricked into thinking that they are coincident for all triangles!). When C is a general convex region, the minima of both quantities are usually at different points in C.

We can also define: Distance ratio(P) as the ratio between longest and shortest distances to boundary of C from P and the Distance difference(P) as difference between these distances. It is seen that even when C is a triangle, these two quantities are minimized at different points in C.

Questions: Among all convex C's of unit diameter, which shape maximizes the distance between the minima of the distance ratio and distance difference? This question can be asked for the {chord ratio, chord difference} pair as well.

Note 1: More basically, when are these centers unique? Indeed, quantities defined above might have more than one minimum point for general C. For example, a square appears to have 4 distinct points in its interior that minimize the chord ratio.

Note 2: In 3D, one can define analogous quantities using (instead of chords) say, areas/perimeters of planar sections of C containing a point P inside C.