This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
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).
- 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?