I have ten meters of mesh fence. I want to make two enclosures for my rabbits who can’t stand each other (they can share a wall). What is the shape and the area of the largest enclosure (in terms of area) I can build? Each rabbit needs to have access to the same area. The shape can be arbitrary (although it would be nice if it were continuous or smooth to some extent, and each area contiguous).
Examples:
A square of 2x2m, divided by a 2m wall. Area: 4 square meters
A circle of radius 1.21m, divided by a wall. Area: ~4.58 square meters.
Is it possible to do better?
Here’s another solution, that is suboptimal but might be preferable to the soap bubble solution in practice: an ellipse with long axis 1.592 and short axis 0.972. This should be close to the optimal ellipse (I used Kepler’s approximate P=pi * (a+b) formula for the perimeter). It gives an area of 4.861 which is close enough to the current optimum, and looks like this: