Understanding the Existence of Extreme Points in Convex Sets

The study of convex sets and their properties is a fundamental aspect of convex analysis, a branch of mathematics with applications in optimization, economics, and geometry. One of the intriguing questions in this field is whether every convex set must have at least one extreme point. This article aims to clarify the conditions under which extreme points exist in convex sets and to explore counterexamples and proofs related to the claim.

Convex Sets and Extreme Points

A set C in a given vector space is convex if for any two points x, y in C, the line segment joining x and y is entirely contained in C. An extreme point of a convex set C is a point that cannot be expressed as a non-trivial convex combination of two other points in C. In simpler words, an extreme point is a boundary point of the set that does not lie in the open relative interior of any line segment lying entirely within C.

Counterexample in Unbounded Sets

To begin with, let us consider the statement: 'Every convex set has at least one extreme point.' This is not correct. To illustrate this, we can use the following counterexample with an unbounded set, specifically, the real number plane, denoted as R2:

Counterexample: The Real Number Plane

The real number plane R2 is a convex set. However, it contains no extreme points. To see why, consider any point (x, y) in R2. This point can be expressed as a convex combination of any other point in R2 with itself, like so:

(x, y)  0.5(x, y)   0.5(x, y)

Since any point in R2 can be written in this manner, it is never an extreme point. Therefore, the real number plane R2 does not have any extreme points.

Counterexample in Bounded Sets

Even in the case of bounded sets, the claim that every convex set must have an extreme point does not hold true. Consider the open unit disk in R2, which is defined as the set of all points (x, y) such that x2 y2 1. This open set is convex, but it contains no extreme points. To show this, note that any point within the open unit disk can always be written as a convex combination of other points within the disk. For instance, let (x, y) be any point in the disk. Then, for any point (x', y') also in the disk, the point (x, y) can be written as:

(x, y)  (1 - t)(x', y')   t(x, y)

for some 0 t 1. Therefore, any point in the open unit disk is not an extreme point.

Conditions for Extreme Points in Convex and Compact Sets

However, if a set is both convex and compact, then it is possible to prove that there will be extreme points. A set is compact if it is closed and bounded. The Extreme Point Theorem (or Krein–Milman theorem) states that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points. This theorem provides a rigorous proof that every convex and compact set must have at least one extreme point. Below is a simplified sketch of the proof:

Simplified Sketch of the Proof

To prove the theorem, we can use the contradiction method. Suppose C is a convex and compact set in ?n and assume, for the sake of contradiction, that C has no extreme points. Then, for every point x in C, there exist distinct points x', x'' in C and a real number 0 t 1 such that:

x  t x'   (1 - t) x''

Since C is compact, we can find a finite subcover of C. By considering the nature of the convex combination and using the properties of closed and bounded sets, it can be shown that the set of all such points x forms a non-empty compact set. However, this leads to a contradiction because, in a compact set, every point is a boundary point without being a convex combination of other points. Hence, our assumption that C has no extreme points must be false, and it must have at least one extreme point.

Conclusion

In summary, the statement that every convex set must have at least one extreme point is not always true. It depends on the nature of the set. For unbounded and open bounded convex sets, the claim does not hold. However, for convex and compact sets, the existence of extreme points is guaranteed by the Extreme Point Theorem. Understanding these nuances is crucial for applications in various fields such as optimization and economics.