In

n dimensional euclidian space **R**^{n}, a

*polytope* is a

compact set delimited by a

finite number of

hyperplanes.

In other words, it's the n dimensional generalisation of a polygon (in **R**^{2}) and a polyhedron (in **R**^{3}). Some authorities prefer to modify the definition above not to allow "holes" in the polytope (in more dimensions, holes become even more complicated than in 3 dimensions!).

Many amazing properties of polytopes, especially convex ones, are known. For instance, linear programming occurs on a polytope (assuming all variables may be bounded); the analysis of complexity for algorithms to solve linear programming involves combinatorial properties of polytopes.