Two Centers of Planar Convex Regions

**Definition:** A line segment with both end points on the boundary of a planar convex region $C$ is called a chord of $C$.

1. Consider any point P within a given planar convex region $C$. From among all chords of $C$ that pass thru P, find that chord for which the ratio between the 2 segments into which P divides the chord (longer seg : shorter seg) is a maximum and note this maximum. Now, how does one find the position of P inside $C$ such that this maximum ratio is a *minimum*?

**Remarks:** Numerical experiments indicate that when $C$ is a triangle, the optimal position of P is always the centroid of the triangle. But for general convex $C$, P may not be at the center of mass.

2. Consider any P within a planar convex region $C$. For every angle of orientation measured from P, there is a unique chord of $C$ that passes thru P. Consider the average over the orientation of the length of the corresponding chord. Now, how does one find that position of P such that the average length of chord thru it is *maximum*?

**Remarks:** Numerically, when $C$ is a triangle, the optimal position of P *does not* seem to be on the centroid or incenter of $C$. Note also that the average length of chord thru P can also be considered over each chord starting at each point on the boundary of $C$ and passing thru P - in this case we might have a different optimal P.

Note (additional question): We can also look for that interior point that minimizes the ratio between the max distance from it to the boundary of C and the min distance from it to the boundary of C.

Update(Feb 10th, 2021): Here is the Mathoverflow discussion page on this question. Thanks to Patrick Schnider. https://mathoverflow.net/questions/383415/on-two-centers-of-convex-regions