I am learning about convex sets and extreme points from a course on linear programming. I came across a theorem that states that every closed convex set has an extreme point if and only if it does not contain a line. I cannot understand what it means for the convex set to contain a line. Can you provide an example of two convex sets that do and do not contain lines?
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1A half plane contains lines, for example, those parallel to the line that define it. A square (boundary and interior) doesn't contain entire lines. The extreme points of the square being the vertices. – NDB Jul 28 '23 at 01:26
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A set $C$ contains a line iff for some $x,y \in C$ and all $t \in \mathbb{R}$ we have $tx+(1-t)y \in C$. The set ${ (x,y) | x \ge 1 }$ contains a line, the set ${ (x,y) | x^2+y^2 \le 1 }$ does not contain a line. – copper.hat Jul 28 '23 at 05:05