Centralness of convex planar regions - alternative definitions

Centralness as defined by Yaglom and Boltyanskii:
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).
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A pair of alternative definitions for centralness of C:
Centralness is the ratio between the area of the largest area centrally symmetric figure that is contained by C to that of C.
Alternatively, centralness can also be defined as the ratio of the area of the smallest centrally symmetric figure that contains C to that of C.
There is no reason for the same shape of C to extremize both these ratios.
Note: We can also consider using perimeter instead of area to define centralness.
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