Convex polytope

A convex polyhedron created from the intersection of various random half-planes

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,[1][2] while others treat convexity as a property of a general polytope.[3] 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 an abstract 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 wasn't proven complete until 1975,[4] the Archimedean solids could've been discovered as early as 200 BCE by Archimedes.[5] Likewise, while the classification of non-convex uniform polychora remains open as of 2020,[6] the convex uniform polychora have been enumerated since 1965.[7] This in part due to the fact that convexity imposes a requirement on the angles meeting at each element (e.g. in a polyhedron, the angles meeting at each vertex can't exceed 2π), while general polytopes have no such requirement.

Examples

All of the following are examples of convex polytopes:

The following are examples of non-convex polytopes:

Definitions

There isn't a standard, agreed upon way to define the property of convexity for a general polytope. Oftentimes, convexity is just determined visually, following the loose description of "a polytope without clefts and self-intersections". The following are proposals for a more rigorous definition:

Convex hulls

Convex polytopes can be defined in terms of convex hulls. The convex hull ${\displaystyle X}$ of a set of points ${\displaystyle S}$ is the smallest convex set containing such points, in the sense that any other convex set that contains the points of ${\displaystyle S}$ also contains the set ${\displaystyle X}$. Intuitively, it can be thought as the shape that an elastic hypersphere (or a rubber band in the 2D case) would take when fit around the points.

The surface of the convex hull of any finite amount of points can be divided into flat faces. These faces form a polytope that is also known as the convex hull of the points. A convex polytope can therefore be defined as follows:

A convex polytope is a polytope equal to the convex hull of its vertices.

Convex hulls of polytopes have various interesting properties:

• The convex hull of a polytope *P* has at least as much symmetry as *P* (though it can have more, see for instance the tetrahemihexahedron).
• The convex hull of an isogonal polytope is isogonal.

Recursive definition

The notion of a convex polytope can be defined recursively, as a polytope satisfying the following four characteristics:

• The interiors of its facets do not intersect one another.
• The interior angles between facets are less than π.
• The polytope is not skew.
• All of its facets are convex.

Interior definition

A convex polytope may be defined as one whose interior has a unique density of 1 and is a convex set.

References

1. Grünbaum, Branko. Kaibel, Volker; Klee, Victor; Ziegler, Günther M. (eds.). Convex Polytopes (2 ed.). p. 51. ISBN 978-0-387-40409-7.
2. Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, 152, Berlin, New York: Springer-Verlag, p. 4
3. Coxeter, Harold S. M. (1963). Regular polytopes (2 ed.). London. pp. 126, 288.
4. Skilling, John (1975). "The complete set of uniform polyhedra". Philosophical Transactions of the Royal Society A. 278 (1278): 111–135. doi:10.1098/rsta.1975.0022.
5. Grünbaum, Branko (2009), "An enduring error", Elemente der Mathematik, 64 (3): 93, doi:10.4171/EM/120, MR 2520469
6. Bowers, Jonathan. "Uniform Polychora and Other Four Dimensional Shapes". Retrieved June 13, 2020.
7. Conway, John H.; Guy, Michael (1965). "Four-Dimensional Archimedean Polytopes". Proceedings of the Colloquium on Convexity at Copenhagen: 38–39.