A convex 3-complex not simplicially isomorphic to a strictly convex complex

Abstract

A set X in euclidean space is convex if the line segment joining any two points of X is in X . If X is convex, every boundary point is on an ( n − 1)-plane which contains X in one of its two closed half-spaces. Such a plane is called a support plane for X . A simplicial complex K in is called strictly convex if | K | (the underlying space of K ) is convex and if, for every simplex σ in ∂ K (the boundary of K ) there is a support plane for | K | whose intersection with | K | is precisely σ In this case | K | is often called a simplicial polytope .