Convexity

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A convex polytope is, loosely speaking, a polytope without clefts, holes, or self-intersections, visually the result of taking a finite set of points in a containing space and enveloping them as tightly as possible in shrink wrap. More formally, convex polytopes may be viewed as point sets, and their defining property is that they are convex sets: given any line segment bounded by two points in the polytope, all points on that line segment are also in the polytope.

Convex polytopes have been extensively studied, much more so than their non-convex counterparts, because they are quite well-behaved compared to general polytopes and have numerous important applications in the sciences. For example, convex polytopes are closely related to linear inequalities and are major objects in the fields of convex geometry and convex optimization. More technically, they are all topologically equivalent to hyperspheres, and they all obey the Euler characteristic and the more general Dehn–Sommerville equations.

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.

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[note 1] 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[edit | edit source]

All of the following are examples of convex polytopes:

The following are examples of non-convex polytopes:

Convex hull[edit | edit source]

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[edit | edit source]

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 (V-polytopes)[edit | edit source]

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.

Intersecting half-spaces definition (H-polytopes)[edit | edit source]

In Euclidean n -space, a convex polytope can also be defined as an intersection of finitely many closed half-n -spaces.

An important result of Weyl and Minkowski is that bounded -polytopes and -polytopes characterize the exact same set of polytopes for every n .

Recursive definition[edit | edit source]

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[edit | edit source]

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

Comparison of definitions[edit | edit source]

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[edit | edit source]

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

Quasi-convexity[edit | edit source]

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[edit | edit source]

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.

Finite isogonal polytopes are all weakly convex.

Specific convex polytope classes[edit | edit source]

See also[edit | edit source]

References[edit | edit source]

  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.

Notes[edit | edit source]

  1. "Strictly convex" has a different definition outside of polytopes, where it may refer to convex sets whose topological boundary contains no nontrivial line segments, and therefore must be curved everywhere. See e.g. Schramm, Oded. "How to cage an egg."

External links[edit | edit source]