A simple polytope is a 
-dimensional
convex polytope with exactly 
edges incident with each
polytope vertex. Equivalently, exactly 
facets meet at each polytope
vertex. Here a facet is a face
of dimension 
.
Its dual is a simplicial polytope (Ziegler
1995). In three dimensions this is the combinatorial definition of a simple
polyhedron, distinct from the geometric usage of that term.
The connection with the simplex is local. For 
, a 
-dimensional convex polytope
is simple iff every vertex figure
is a 
-simplex (Renner 2009). The simplicial condition instead
requires every facet to be a simplex.
Thus a simple polytope has simplices
as vertex figures, while a simplicial
polytope has simplices as facets
(Fukuda 2004).
Every convex polygon is simple. The cube is simple, since three edges meet at each polytope
vertex. The octahedron is not simple, since four
meet at each polytope vertex. An 
-simplex is both a simple polytope
and a simplicial polytope. The cube
shows that a simple polytope need not be a simplex.
The face counts satisfy the Dehn-Sommerville equations. Partitioned permutohedra are another family of simple polytopes (Horiguchi et al. 2026).
