# On comparing planar convex regions of equal perimeter and area

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)