Convex polytope

A convex polytope is, loosely speaking, a polytope without clefts and without self-intersections. Convex polytopes represent convex sets, sets of points such that every point in any dyad defined by a pair of points in the set is also in the set. Similarly to the notion of a polytope itself, there are various similar but distinct notions of convex polytopes: some authors define all polytopes as convex point sets, thus making every polytope automatically convex, while others treat convexity as a property of a general polytope. This article deals with the latter notion.

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. This has lead to some authors to use the words "convex polytope" and "polytope" interchangeably.

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 wasn't 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 have been enumerated since 1965.