**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:**

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?

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)