In mathematics, an abstract polytope is a combinatorial structure with properties similar to those shared by a more classical polytope. Abstract polytopes include the polygons, the platonic solids and other polyhedra, tesselations of the plane and higher-dimensional spaces, and of other manifolds such as the torus or projective plane, and many other objects (such as the 11-cell and the 57-cell) that don't fit well into any "normal" space.

More precisely, an abstract polytope is a set of objects, supposed to represent the vertices, edges and so on — the faces — of the polytope. An "order" is imposed on the set.

In the study of optimization, linear programming studies the maxima and minima of linear functions constricted to the boundary of an n-dimensional polytope.

Examples

• The tesseract is one of the simplest higher dimensional polytopes.
• A simplex is a polytope.