Give example of a set which has No Extreme Point !!..

Question

Give example of a set in $\mathbb R^2$, which has no extreme point ??

We were given this question for assignment !!..I thought of a simple line but doing some research I stumbled upon this solution which said,

$$S=\{(x,y)\in\mathbb R^{2}: y \geq 0\}$$

But doest this implies that every point on $y$ axis can be treated as an extreme point in itself...so it has infinite points rather than no extreme point !!..I am unable to understand this extreme point concept...

What is an extreme point ??...I follow this definition: an extreme point is a point in a convex set $K$ that is not an interior point of any line segment lying entirely in $K$.

So if i follow the above definition and solution from the book, doesnt all the points on $y$ axis and any point in upper half plan of $\mathbb R^2$ satisfy this extreme point condition?

Answer 1

Definition : An extreme point $x_e$ of a convex set $\mathcal{C}$ is a point, belonging to its closure $\overline{\mathcal{C}}$, that is not expressible as a convex combination of points in $\overline{\mathcal{C}}$ distinct from $x_e$ ;

i.e, for $x_e\in \overline{\mathcal{C}}$ and all $x_1 , x_2 \in \mathcal{C} \setminus \{x_e\}$

$$\lambda x_1 + (1 − \lambda)x_2 \neq x_e , \ \ \ \ \lambda \in [0, > 1] $$

Point A is extreme; Point B is extreme, Point C is extreme too, but D is not

You can see that all point in the boundary of your set are like D.

Actually there is a theorem says :

Theorem : A nonempty closed convex set containing no lines has at least one extreme point.

So if you are looking for set without extreme point, you have to look at convex sets containing at least one line. This is consistent with your set.

Reference : [J. Dattorro] Convex Optimization & Euclidean Distance Geometry