The basic question is to find that planar convex region for which diameter is a maximum when area and perimeter are specified.

A partial answer is given here: http://nandacumar.blogspot.com/2012/11/maximizing-and-minimizing-diameter-ii.html which goes: If the specified perimeter is not more than that of the Reuleaux triangle with the specified area, then, the required planar convex region is a region of constant width. The following question remains, as was noted in the above-linked page:

Question: For specified area $A$ and perimeter $P$ where $P$ is greater than the perimeter of the Reuleaux triangle of area $A$, which planar convex shape maximizes diameter?

When the specified $P$ is steadily increased keeping $A$ fixed, will an ellipse become the answer at any stage?

Note: higher dimensional analogs to the questions could also be considered.

The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.

A partial answer is given here: http://nandacumar.blogspot.com/2012/11/maximizing-and-minimizing-diameter-ii.html which goes: If the specified perimeter is not more than that of the Reuleaux triangle with the specified area, then, the required planar convex region is a region of constant width. The following question remains, as was noted in the above-linked page:

Question: For specified area $A$ and perimeter $P$ where $P$ is greater than the perimeter of the Reuleaux triangle of area $A$, which planar convex shape minimizes diameter?

When the specified $P$ is steadily increased keeping $A$ fixed, will an ellipse become the answer at any stage?

Note: higher dimensional analogs to the questions could also be considered.