[math-fun] Turning the Tables

There's a famous (and easy, once you've seen the idea) theorem: If you have a rectangular table with one leg in each corner, and an arbitrary (smooth) uneven floor, you can always find a way of placing the table so that all four feet touch the floor. If instead your table is triangular, with a leg in each corner and one somewhere in the middle, then it's easy to make a floor for which no position puts all the feet on the floor. (E.g., height = x^2+y^2.) What happens for arbitrary convex quadrilaterals? (Note: I have elided a few details above; e.g., if your floor is finite and smaller than the table then there's nowhere to put it :-).) -- g