Extreme point

In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or corner point of S.[1]

For other uses, see Extreme point (disambiguation).
A convex set in light blue, and its extreme points in red.

Definition

Throughout, it is assumed that S is a real or complex vector space.

For any x, x1, x2 ∈ S, say that x lies between[2] x1 and x2 if x1 ≠ x2 and there exists a 0 < t < 1 such that x = tx1 + (1 − t)x2.

If K is a subset of X and x ∈ K, then x is called an extreme point[2] of K if it does not lie between any two distinct points of K. That is, if there does not exist x1, x2 ∈ K and 0 < t < 1 such that x1 ≠ x2 and x = tx1 + (1 − t) x2. The set of all extreme points of K is denoted by extreme(K).

Characterizations

The midpoint[2] of two elements x and y in a vector space is the vector 1/2(x + y).

For any elements x and y in a vector space, the set [x, y] := {tx + (1 − t)y : 0 ≤ t ≤ 1} is called the closed line segment or closed interval between x and y. The open line segment or open interval between x and y is (x, x) := ∅ when x = y while it is (x, y) := {tx + (1 − t)y : 0 < t < 1} when x ≠ y.[2] The points x and y are called the endpoints of these interval. An interval is said to be non-degenerate or proper if its endpoints are distinct. The midpoint of an interval is the midpoint of its endpoints.

Note that [x, y] is equal to the convex hull of {x, y} so if K is convex and x, y ∈ K, then [x, y] ⊆ K.

If K is a nonempty subset of X and F is a nonempty subset of K, then F is called a face[2] of K if whenever a point p ∈ F lies between two points of K, then those two points necessarily belong to F.

Theorem[2] — Let K be a non-empty convex subset of a vector space X and let p ∈ K. Then the following are equivalent:

  1. p is an extreme point of K;
  2. K ∖ { p } is convex;
  3. p is not the midpoint of a non-degenerate line segment contained in K;
  4. for any x, y ∈ K, if p ∈ [x, y] then x = p or y = p;
  5. if x ∈ X is such that both p + x and p − x belong to K, then x = 0;
  6. { p } is a face of K.

Examples
  • If a < b are two real numbers then a and b are extreme points of the interval [a, b]. However, the open interval (a, b) has no extreme points.[2]
  • An injective linear map F : X → Y sends the extreme points of a convex set C ⊆ X to the extreme points of the convex set F(C).[2] This is also true for injective affine maps.
  • The perimeter of any convex polygon in the plane is a face of that polygon.[2]
  • The vertices of any convex polygon in the plane ℝ2 are the extreme points of that polygon.
  • The extreme points of the closed unit disk in ℝ2 is the unit circle.
  • Any open interval in ℝ has no extreme points while any non-degenerate closed interval not equal to ℝ does have extreme points (i.e. the closed interval's endpoint(s)). More generally, any open subset of finite-dimensional Euclidean space ℝn has no extreme points.

Properties

The extreme points of a compact convex form a Baire space (with the subspace topology) but this set may fail to be closed in X.[2]

Theorems

Krein–Milman theorem

The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points.

Krein–Milman theorem — If S is convex and compact in a locally convex space, then S is the closed convex hull of its extreme points: In particular, such a set has extreme points.

For Banach spaces

These theorems are for Banach spaces with the Radon–Nikodym property.

A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonempty closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded).[3]

Theorem (Gerald Edgar) — Let E be a Banach space with the Radon-Nikodym property, let C be a separable, closed, bounded, convex subset of E, and let a be a point in C. Then there is a probability measure p on the universally measurable sets in C such that a is the barycenter of p, and the set of extreme points of C has p-measure 1.[4]

Edgar's theorem implies Lindenstrauss's theorem.

k-extreme points

More generally, a point in a convex set S is k-extreme if it lies in the interior of a k-dimensional convex set within S, but not a k+1-dimensional convex set within S. Thus, an extreme point is also a 0-extreme point. If S is a polytope, then the k-extreme points are exactly the interior points of the k-dimensional faces of S. More generally, for any convex set S, the k-extreme points are partitioned into k-dimensional open faces.

The finite-dimensional Krein-Milman theorem, which is due to Minkowski, can be quickly proved using the concept of k-extreme points. If S is closed, bounded, and n-dimensional, and if p is a point in S, then p is k-extreme for some k < n. The theorem asserts that p is a convex combination of extreme points. If k = 0, then it's trivially true. Otherwise p lies on a line segment in S which can be maximally extended (because S is closed and bounded). If the endpoints of the segment are q and r, then their extreme rank must be less than that of p, and the theorem follows by induction.

See also

Citations
    1. Saltzman, Matthew. "What is the difference between corner points and extreme points in linear programming problems?".
    2. Narici & Beckenstein 2011, pp. 275-339.
    3. Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562.
    4. Edgar GA. A noncompact Choquet theorem. Proceedings of the American Mathematical Society. 1975;49(2):354-8.

    Bibliography
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