math.MG — Metric Geometry

A counterexample to the table conjecture on the sphere

Contributed by Pablo Soberón

We disprove Matschke's table conjecture, which asserts that every continuous real-valued function on $S^2$ takes a common value on the vertices of some rotated copy of every prescribed inscribed square. The counterexample is the cubic polynomial $H(x,y,z)=14z^3-6z+y+xz$ and a square contained in a plane at distance $1/\sqrt{7}$ from the origin. We give an elementary proof.