Subpolytopes of Cyclic Polytopes

A remarkable result of Shemer states that the combinatorial structure of a neighbourly 2 m -polytope determines the combinatorial structure of each of its subpolytopes. From this, it follows that every subpolytope of a cyclic 2 m -polytope is cyclic. In this note, we present a direct proof of this consequence that also yields that certain subpolytopes of a cyclic (2 m + 1)-polytope are cyclic.