In 1970 Roger Fenn
showed that one could always balance a square table on a hill. Soon after,
Joseph Zaks stated that one can always translate a triangular chair to balance
on a hill. Recent results have been found by E. H. Kronheimer and P. B.
Kronheimer. This paper gives precise statements of the theorems and shows
that:
Compact support for the hill in the Chair Theorem cannot be replaced by
lim|x|→∞f(x) = 0 (answering a question of Zaks).
The isometry of the Table cannot be replaced by a translation.
The square Table cannot be replaced by an n-gon table for n ≥ 5.
The Table Theorem is still open for cyclic quadrilaterals.
