'Centralness Coefficient' Etc.

There has been a shift of focus to MathOverflow: https://mathoverflow.net/users/142600/nandakumar-r

However, some discussions there seemed worth collecting in this post.

1. The recent encounter with centralness coefficient was important enough to copy here as well.
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https://mathoverflow.net/questions/396579/a-center-of-convex-planar-regions-based-on-chords
A point P in the interior of a planar convex region C divides every chord of C that passes thru it into 2 segments. Consider, for each chord thru P, the ratio between the length of the shorter segment and length of the longer segment.

For every P, this 'chord length ratio' has the maximum value 1 (for every P, there is at least one chord of C for which P is the mid point) but its least value varies with P. That position of P where the least value of this ratio is maximum (in other words, position of P where the values of the chord length ratio are within narrowest bounds) can be called a center of C and the highest value of the least chord length ratio over all interior points can be called the centralness coefficient of C.

It can be shown that the centralness coefficient of any convex figure cannot be less than 1/2 (this value holds for all triangles) and at least 3 chords pass thru a center of C with chord length ratio equal to the centralness coefficient (e.g. the medians of a triangle).

Question: Is the center of any convex region a unique point?
Remarks: If C is centrally symmetric, the center is unique. It appears to coincide with the center of mass even for regular polygons with odd number of sides. One can also ask about the relationship of this center with other special points such as center of mass — how far apart they could possibly be etc. How does one algorithmically determine the center(s) of a convex n-gon?
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2. The centralness coefficient is unlike area or perimeter or diameter a quantity that does not change with scaling - it is about the shape of the convex region. So, one could naturally bring it in as among the quantities to be equalized when a convex region is partitioned into n pieces and see the implications.
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3. Another question that was recently pondered was: how many quantities need to be equalized among the convex faces of a convex polyhedron to guarantee that all the faces are mutually congruent? It has been argued here: https://mathoverflow.net/questions/395013/convex-polyhedra-with-non-congruent-faces that with only area and perimeter equal among faces, all faces of a convex polyhedron can be mutually non-congruent.

However, the question can be raised: can one constrain a few more quantities to be equal among faces to guarantee, say, "at least a pair of faces will be congruent" or "all faces are necessarily congruent"?
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There should be updates to this post as I learn more about these questions...