Convex polytope

A convex set or convex region is a set of points such that for every pair of points on the set, every point on the line segment connecting both points is also on the set. A convex polytope is a polytope that has an interior with a single density of 1 and is a convex set. Equivalently, convex polytopes are those that can be obtained as the convex hull of a finite set of points, or those whose facets correspond to the planes of the surface of the intersection of a finite amount of half-planes.

Historically, convex polytopes have been more expansively studied than their non-convex counterparts. In various senses, convex polytopes are better behaved: they all obey the Euler characteristic (which non-convex polytopes generally don't), and are all orientable and tame. Also, convex polytopes appear in a wider variety of contexts, such as in the resolution of linear inequalities or as objects of study in convex geometry, while non-convex polytopes are almost always studied from a combinatorial point of view.

Furthermore, convex polytopes of various classes lend themselves to enumeration more easily. For example, though the enumeration of uniform polyhedra wasn't completed until 1953, and proven complete until 1975, the Archimedean solids could've been discovered as early as 200 BCE by Archimedes. Likewise, while the classification of non-convex uniform polychora remains open as of 2020, the convex uniform polychora were enumerated since 1965.