Non-starshaped spheres (Q1203575)

A simplicial \((d-1)\)-sphere \(S\) is a finite simplicial complex in Euclidean space \(\mathbb{E}^ n\) whose underlying point set is a topological sphere. \(S\) is called starshaped when it is isomorphic to some \(S' \subseteq \mathbb{E}^ d\) bounding a \(d\)-ball which is starshaped with respect to one of its interior points. This last is taken in a strong sense: if \(x\) is the interior point and \(y\) is in \(S'\), then the closed segment joining \(x\) and \(y\) meets \(S'\) in \(y\) alone. In this paper the authors construct \((d-1)\)-spheres which cannot be embedded in \(\mathbb{E}^ d\) as starshaped sets. More precisely they show that, for any \(d \geq 4\), there are non-starshaped simplicial \((d-1)\)- spheres in \(\mathbb{E}^ d\) which have \(d+8\) vertices. They develop consequences which shed some light on the classification of toric varieties in algebraic geometry. Some text seems to be missing at the end of the first paragraph.