Leftovers from the Ham Sandwich Theorem

The traditional proofisa clev eruseoftheBorsuk-Ulamtheorem;see[1,p.120] . As Munkresremarks[5,p.405] ,theham sandwic h theoremisnotelemen tary ,evenindimension2,whereitisknown asthebabyham sandwichtheorem. But isitreally sohard? Consider thefollo wing:\aplanethroughthecentresofgravit y ofeach ofthebodieswill dothetric k" [3,p.57].Unfortunately ,thiseductiv eargumentjustdoesn’t work:a plane throughthecentreofgravit y rarely divides a bodyintohalvesofequalvolume.Forexample,consider a region inR boundedby anequilateral triangle. Herethecentreofgravit y isthecentroidbutitisclear thata linethroughthecentroidparallel toan edgedoesnot divide thetriangle intohalvesofequalarea(seeFigure1).Thickeningthetriangle a little givesa 3-dimensional example.