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Definitions:

  • The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
  • Given two planar convex regions $C_1$ and $C_2$ both with unit perimeter, we define the difference between $C_1$ and $C_2$ as the least value of Hausdorff distance between $C_1$ and $C_2$ can have when the regions are placed above one another and adjusted to minimise the Hausdorff distance between them.

Questions:

  1. What are the specific pair of unit perimeter regions $\{C_1, C_2\}$ with some equal specified area such that the difference between $C_1$ and $C_2$ is maximum?

  2. What are the specific pair of unit perimeter regions $\{C_1, C_2\}$ with equal specified area and equal specified diameter (diameter of a region is the greatest distance between any two points in the region). such that the difference between $C_1$ and $C_2$ is maximum?

(further versions of the question with $C_1$ and $C_2$ sharing equal values of more global quantities – and also higher dimensional versions – are natural)

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We suggest an answer to question 1 above, without proof of optimality:

First we note two facts. Please see https://nandacumar.blogspot.com/2012/11/maximizing-and-minimizing-diameter-ii.html for some more details.

  • For specified values of area and perimeter, the convex region with maximum diameter is formed with two equal circular arcs (how one draws a convex lens).
  • For specified area and perimeter, diameter is minimized by a convex closed curve of constant width.

So choosing these 2 figures as the pair, we get at least a difference of (max diameter - min diameter)/2 when the two figures are superimposed.

Note: When the area specified (with perimeter fixed at 1) is less than that of the Reuleux triangle of unit perimeter, we don't have a convex figure of constant width.

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