A compact convex subset of Euclidean space that is the convex hull of finitely many points; equivalently a bounded intersection of finitely many supporting half-spaces, with a finite face lattice.

Principle

Finite combinatorial data (vertices, edges, faces) together with convexity determine geometry: Carathéodory, Minkowski and supporting hyperplane theorems link combinatorics, metrics and duality.

Demonstration

A cube is the convex hull of its eight vertices and the intersection of six supporting half-spaces. In general, a polytope's faces are themselves polytopes and may be triangulated into simplices.

Misapplication

Calling any bounded set with a polygonal boundary a convex polytope despite nonconvex indentations, or confusing polytopes with infinite convex sets (cones, cylinders) that are not compact.

Consequence

Convex polytopes admit finite descriptions, dual polytopes, f-vector invariants, and algorithmic treatments: linear programming, face enumeration, and volume computation all exploit polytope structure.

Reversal

Dropping boundedness or finiteness of supporting hyperplanes yields unbounded polyhedra or general convex sets with qualitatively different behavior (no finite face lattice, different duality).

Boundary

Definition requires compactness and finite generation by points; it excludes nonconvex polytopes, infinite polytopes (polyhedra), and curved polytopes; topological or combinatorial polytopes without an embedding are separate notions.

Semantic Tension

Tension appears between the geometric (embedded) polytope and abstract polytopes (incidence structure), and between combinatorial invariants (f-vectors) and metric properties (volumes, angles).

Synthesis

A convex polytope is a finitely generated, compact convex body whose finite combinatorial structure (faces, vertices) governs both discrete invariants and continuous geometric properties, enabling duality and algorithmic analysis.