This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
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.