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

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?

• $(0,y)= \bigl( (0,y+1) +(0,y-1)\bigr)/2$. Commented Oct 5, 2014 at 10:24
• If you have difficulties understanding the concept of an extreme point, I suggest you to add two things in your question: (1) The definition of an extreme point that you were given. (2) Your explanation in your own words what it means and what seem to be the extreme points of $S$. With this information it would be much easier to help you. Commented Oct 5, 2014 at 10:31
• sir, i added what you asked me to do.. Commented Oct 5, 2014 at 10:39
• Every point $(x,y)$ lies in the interior of a line segment lying in the set you mention. Lots of line segments. Try picking a point and drawing one. Commented Oct 5, 2014 at 10:43
• Give an example of a question without two exclamation points and two periods (points)
– user12802
Commented Oct 5, 2014 at 23:19

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.

• Can you provide an example of a convex set that contains at least one line? The OP's set Commented Jul 28, 2023 at 0:56