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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).

Questions:

  1. 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.

  1. How does one algorithmically determine the center(s) of a convex $n$-gon?
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    $\begingroup$ I think my answer to a previous question of yours should imply that the center is unique: mathoverflow.net/a/383528/143513 $\endgroup$ Jul 5, 2021 at 16:30
  • $\begingroup$ Thank you very much for pointing this out. In fact, the definition in this post of center as maximizing a minimum and the definition of the special point in the earlier discussion where the question was on minimizing a maximum both appear equivalent. Your nesting argument in mathoverflow.net/a/383528/143513 implies uniqueness of the center at least in 2D. And I just learned in 3D, the 'center' is NOT always a unique point - for a triangular prism with height h, any point in the middle 1/3rd of the segment formed by the centroids of its triangular sections is a center. $\endgroup$ Jul 9, 2021 at 11:44
  • $\begingroup$ An example where Center of a convex region is NOT the center of mass: Consider a circular disk D with center C and a point P a little outside D. Attach to D (1) the region bounded by the tangents to D from P (2) a thin layer to half of the boundary of D on the side opposite to P s.t. center of mass of the total object (D') is again C (area of the thin layer must be > portion between tangents from P). Now, for D', the most 'lopsided' chord thru C is unique (starts from P). As per 'Convex Figures' (6-3), there are at least 3 such most lopsided chords thru the center; so C is not center of D'. $\endgroup$ Jan 1 at 7:13

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