What's the maximum area shape that will fit around a right-angle corner?

This problem is unsolved.  A simple solution called the "Shephard
piano" has area 2.2074+, but this can be improved upon with local
modifications.  A solution exists with area 2.215649+.  It is known
that a maximum area exists, but not whether the shape achieving it is
symmetric, smooth, or even unique.

See Problem G5 in Croft, Falconer, and Guy, _Unsolved Porblems in
Geometry_, Springer-Verlag, 1991.