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).
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 simple and simplicial.
The face counts satisfy the Dehn-Sommerville equations. Partitioned permutohedra are another family of simple polytopes (Horiguchi et al. 2026).
