Convex set

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Illustration of a convex set which looks somewhat like a deformed circle. The (black) line segment joining points x and y lies completely within the (green) set. Since this is true for any points x and y within the set that we might choose, the set is convex.
Illustration of a non-convex set which looks somewhat like a boomerang. Since the red part of the (black and red) line-segment joining the points x and y lies outside of the (green) set, the set is non-convex.

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins the pair of points is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. A convex curve forms the boundary of a convex set.

The notion of a convex set can be generalized to other spaces as described below.

In vector spaces

A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.

Let Template:Mvar be a vector space over the real numbers, or, more generally, some ordered field. This includes Euclidean spaces. A set Template:Mvar in Template:Mvar is said to be convex if, for all Template:Mvar and Template:Mvar in Template:Mvar and all Template:Mvar in the interval [0, 1], the point (1 − t)x + ty also belongs to Template:Mvar. In other words, every point on the line segment connecting Template:Mvar and Template:Mvar is in Template:Mvar. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected. Furthermore, Template:Mvar is strictly convex if every point on the line segment connecting Template:Mvar and Template:Mvar other than the endpoints is inside the interior of Template:Mvar.

A set Template:Mvar is called absolutely convex if it is convex and balanced.

The convex subsets of R (the set of real numbers) are simply the intervals of R. Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot polyhedra are examples of non-convex sets.

Non-convex set

"Concave set" redirects here.

A set that is not convex is called a non-convex set. A polygon that is not a convex polygon is sometimes called a concave polygon,[1] and some sources more generally use the term concave set to mean a non-convex set,[2] but most authorities proscribe this usage.[3][4]

Properties

If Template:Mvar is a convex set in Template:Mvar-dimensional space, then for any collection of Template:Mvar, r > 1, Template:Mvar-dimensional vectors u1, ..., ur in Template:Mvar, and for any nonnegative numbers λ1, ..., λr such that λ1 + ... + λr = 1, then one has:

A vector of this type is known as a convex combination of u1, ..., ur.

Intersections and unions

The collection of convex subsets of a vector space has the following properties:[5][6]

  1. The empty set and the whole vector-space are convex.
  2. The intersection of any collection of convex sets is convex.
  3. The union of a non-decreasing sequence of convex subsets is a convex set. For the preceding property of unions of non-decreasing sequences of convex sets, the restriction to nested sets is important: The union of two convex sets need not be convex.

Closed convex sets

Closed convex sets can be characterised as the intersections of closed half-spaces (sets of point in space that lie on and to one side of a hyperplane).

From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set Template:Mvar and point Template:Mvar outside it, there is a closed half-space Template:Mvar that contains Template:Mvar and not Template:Mvar. The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis.

Convex sets and rectangles

Let C be a convex body in the plane. We can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C. The positive homothety ratio is at most 2 and:[7]

Convex hulls and Minkowski sums

Convex hulls

{{#invoke:main|main}} Every subset Template:Mvar of the vector space is contained within a smallest convex set (called the convex hull of Template:Mvar), namely the intersection of all convex sets containing Template:Mvar. The convex-hull operator Conv() has the characteristic properties of a hull operator:

extensive S ⊆ Conv(S),
non-decreasing S ⊆ T implies that Conv(S) ⊆ Conv(T), and
idempotent Conv(Conv(S)) = Conv(S).

The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets

Conv(S) ∨ Conv(T) = Conv(S ∪ T) = Conv(Conv(S) ∪ Conv(T)).

The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice.

Minkowski addition

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Three squares are shown in the nonnegative quadrant of the Cartesian plane. The square Q1 = [0, 1] × [0, 1] is green. The square {{{1}}}.
Minkowski addition of sets. The sum of the squares Q1=[0,1]2 and Q2=[1,2]2 is the square Q1+Q2=[1,3]2.

In a real vector-space, the Minkowski sum of two (non-empty) sets, S1 and S2, is defined to be the set S1 + S2 formed by the addition of vectors element-wise from the summand-sets

S1 + S2 = {x1 + x2 : x1 ∈ S1, x2 ∈ S2} .

More generally, the Minkowski sum of a finite family of (non-empty) sets Sn is the set formed by element-wise addition of vectors

For Minkowski addition, the zero set {0} containing only the zero vector 0 has special importance: For every non-empty subset S of a vector space

S + {0} = S;

in algebraic terminology, {0} is the identity element of Minkowski addition (on the collection of non-empty sets).[8]

Convex hulls of Minkowski sums

Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:

Let S1, S2 be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls

Conv(S1 + S2) = Conv(S1) + Conv(S2).

This result holds more generally for each finite collection of non-empty sets:

In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.[9][10]

Minkowski sums of convex sets

The Minkowski sum of two compact convex sets is compact. the sum of a compact convex set and a closed convex set is closed.[11]

Generalizations and extensions for convexity

The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets.

Star-convex sets

{{#invoke:main|main}} Let Template:Mvar be a set in a real or complex vector space. Template:Mvar is star convex if there exists an x0 in Template:Mvar such that the line segment from x0 to any point Template:Mvar in Template:Mvar is contained in Template:Mvar. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.

Orthogonal convexity

{{#invoke:main|main}} An example of generalized convexity is orthogonal convexity.[12]

A set Template:Mvar in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of Template:Mvar lies totally within Template:Mvar. It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are valid as well.

Non-Euclidean geometry

The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set.

Order topology

Convexity can be extended for a space Template:Mvar endowed with the order topology, using the total order < of the space.[13]

Let YX. The subspace Template:Mvar is a convex set if for each pair of points a, b in Template:Mvar such that a < b, the interval (a, b) = {xX : a < x < b} is contained in Template:Mvar. That is, Template:Mvar is convex if and only if for all a, b in Template:Mvar, a < b implies (a, b) ⊆ Y.

Convexity spaces

The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms.

Given a set Template:Mvar, a convexity over Template:Mvar is a collection 𝒞 of subsets of Template:Mvar satisfying the following axioms:[14][5][6]

  1. The empty set and Template:Mvar are in 𝒞
  2. The intersection of any collection from 𝒞 is in 𝒞.
  3. The union of a chain (with respect to the inclusion relation) of elements of 𝒞 is in 𝒞.

The elements of 𝒞 are called convex sets and the pair (X, 𝒞) is called a convexity space. For the ordinary convexity, the first two axioms hold, and the third one is trivial.

For an alternative definition of abstract convexity, more suited to discrete geometry, see the convex geometries associated with antimatroids.

See also

References

  1. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  2. Weisstein, Eric W., "Concave", MathWorld.
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}
  4. {{#invoke:citation/CS1|citation |CitationClass=citation }}
  5. 5.0 5.1 Soltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian).
  6. 6.0 6.1 {{#invoke:citation/CS1|citation |CitationClass=book }}
  7. Template:Cite doi
  8. The empty set is important in Minkowski addition, because the empty set annihilates every other subset: For every subset Template:Mvar of a vector space, its sum with the empty set is empty: S + ∅ = ∅.
  9. Theorem 3 (pages 562–563): Template:Cite news
  10. For the commutativity of Minkowski addition and convexification, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the convex hulls of Minkowski sumsets in its "Chapter 3 Minkowski addition" (pages 126–196): {{#invoke:citation/CS1|citation |CitationClass=book }}
  11. Lemma 5.3: {{#invoke:citation/CS1|citation |CitationClass=book }}
  12. Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in: Computational Morphology, 137-152. Elsevier, 1988.
  13. Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
  14. {{#invoke:citation/CS1|citation |CitationClass=book }}

External links

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