Convex polytope

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

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 line segment 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.

In various senses, convex polytopes are better behaved, in contrast to general non-convex polytopes. Combinatorially, they all obey the Euler characteristic and the more general Dehn–Sommerville equations. Moreso, and are all orientable and tame.

Historically, convex polytopes have been more expansively studied than their non-convex counterparts. They find wider use in mathematics, 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. 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 2023,[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.

A polytope is strictly convex if it is convex and no two facets are in the same hyperplane. This additional requirement is used in the definitions of the Johnson solids, Blind polytopes, and CRF polytopes to restrict the possibilities.

Examples

All of the following are examples of convex polytopes:

The following are examples of non-convex polytopes:

Convex hull

The convex hull X of a set of points S is the smallest convex set containing such points, in the sense that any other convex set that contains the points of S also contains the set 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.

The convex hull of a polytopes vertices is also simply called the convex hull of that polytope. 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.

Definitions

There isn't a single 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 more rigorous definitions used by various authors:

Convex hull definition

Convex polytopes can be defined in terms of the convex hull as follows:

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

Alternatively a polytope is convex if it is the convex hull of a locally-finite set of points.

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.

Comparison of definitions

This polyhedron with 24 square faces is convex by the interior definition but not by the convex hull definition or the recursive definition.

While the definitions presented express the same loose idea they are not strictly equivalent. The interior definition allows polyhedra with coplanar faces to be convex, while the recursive definition and the convex hull definition do not allow these polytopes to be convex.

Related notions

In addition to convexity and strict convexity, there are a number of other weaker forms of convexity employed in different contexts.

Quasi-convexity

A polyhedron, 𝓟, is quasi-convex if all the edges of its convex hull are edges of 𝓟. This definition was introduced by Norman Johnson for the study of Stewart toroids. All convex polyhedra are quasi-convex.

Weak convexity

Jessen's icosahedron is weakly convex but not convex.

A polytope is weakly convex if all of its vertices are vertices of its convex hull. All strictly convex polytopes are weakly convex, but some convex polytopes with coplanar facets are not weakly convex.

See also

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 March 4, 2023.
  7. Conway, John H.; Guy, Michael (1965). "Four-Dimensional Archimedean Polytopes". Proceedings of the Colloquium on Convexity at Copenhagen: 38–39.

External links