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What is simple/simplicial polytope?

A $ d$-polytope is called simple if each vertex is contained in exactly $ d$ facets. A $ d$-polytope is called simplicial if each facet contains exactly $ d$ vertices. By definition, a dual of simple (simlicial) polytope is simplicial (simple, respectively). Every facet of a simplicial $ d$-polytope is a $ (d-1)$-simplex. Each vertex of a simple $ d$-polytope is contained in exactly $ d$-edges.

A $ d$-cube is a simple polytope and a $ d$-simplex is both simple and simplicial.

Komei Fukuda 2004-08-26