In
n dimensional euclidian space Rn, 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 R2) and a polyhedron (in R3). 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.