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For a planar convex region $C$ and an interior point $P$ we define:

  • the centralness ratio at $P$ is $$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\text{ is a chord of }C\text{ through }P\right)$$

  • the depth ratio at $P$ is: $$\frac{\text{distance from }P\text{ to closest point on boundary}}{\text{distance from }P\text{ to farthest point on boundary}}$$

Both ratios go to zero at the boundary of $C$. Here, we ask about averages of these ratios over regions.

Questions: Among all convex planar regions of unit area and specified

  • (1) perimeter or (2) diameter,

over which region is the average value of the

  • (a) centralness ratio or (b) depth ratio
  • maximized or minimized?

Remarks: A 'global' centralness ratio can also be defined for $C$ as a whole as the maximum value of this ratio over all P in the interior of $C$. It is known that this global ratio is a minimum (1/2) if and only if $C$ is any triangle (links below). And if the 'global' depth ratio of $C$ is defined as its maximum value over $C$ as a whole, it is easily seen to have no global minimum (the ratio goes to 0 for degenerate $C$). However, with both area and perimeter of C specified, the global depth ratio seems to have a well-defined maximum (guess: when $C$ is an ellipse) and minimum.

Indeed, with area and perimeter specified, a triangle can be the shape minimizing the global centralness ratio only for a certain range of perimeter values above a critical value - for lower perimeter values, there seem to be a range of shapes that terminate in the circular disk.

Above considerations seem to lead to a question: is the question of optimizing average values of centralness and depth ratios more meaningful than 'constrained' versions (like the area and perimeter specified) of the global centralness and depth ratios?

Some related posts are:

  1. On two centers of convex regions
  2. A center of convex planar regions based on chords
  3. A ratio to measure 'roundedness' of planar convex regions

Note: This post was written with K Sheshadri.

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