Definition: A line segment with both end points on the boundary of a planar convex region $C$ is called a chord of $C$.
- 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 and characterize 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.
- Consider any $P$ within a given 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 and characterize 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 1: Both questions above appear to have natural higher dimensional analogs.
Note 2: (this is actually, a further 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. Numerically, when C is a triangle, this 'center' is not seen to coincide with centroid or incenter.