Editing Extreme point (section)

===Characterizations===

The '''{{visible anchor|midpoint}}'''{{sfn|Narici|Beckenstein|2011|pp=275-339}} of two elements <math>x</math> and <math>y</math> in a vector space is the vector <math>\tfrac{1}{2}(x+y).</math>

For any elements <math>x</math> and <math>y</math> in a vector space, the set <math>[x, y] = \{t x + (1-t) y : 0 \leq t \leq 1\}</math> is called the '''{{visible anchor|closed line segment}}''' or '''{{visible anchor|closed interval}}''' between <math>x</math> and <math>y.</math> The '''{{visible anchor|open line segment}}''' or '''{{visible anchor|open interval}}''' between <math>x</math> and <math>y</math> is <math>(x, x) = \varnothing</math> when <math>x = y</math> while it is <math>(x, y) = \{t x + (1-t) y : 0 < t < 1\}</math> when <math>x \neq y.</math>{{sfn|Narici|Beckenstein|2011|pp=275-339}} The points <math>x</math> and <math>y</math> are called the '''{{visible anchor|endpoints}}''' of these interval. An interval is said to be a '''{{visible anchor|non−degenerate interval}}''' or a '''{{visible anchor|proper interval}}''' if its endpoints are distinct. The '''{{visible anchor|midpoint of an interval}}''' is the midpoint of its endpoints. 

The closed interval <math>[x, y]</math> is equal to the [[convex hull]] of <math>(x, y)</math> if (and only if) <math>x \neq y.</math> So if <math>K</math> is convex and <math>x, y \in K,</math> then <math>[x, y] \subseteq K.</math> 

If <math>K</math> is a nonempty subset of <math>X</math> and <math>F</math> is a nonempty subset of <math>K,</math> then <math>F</math> is called a '''{{visible anchor|face}}'''{{sfn|Narici|Beckenstein|2011|pp=275-339}} of <math>K</math> if whenever a point <math>p \in F</math> lies between two points of <math>K,</math> then those two points necessarily belong to <math>F.</math>

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=275-339}}|math_statement=
Let <math>K</math> be a non-empty convex subset of a vector space <math>X</math> and let <math>p \in K.</math> 
Then the following statements are equivalent:
<ol>
<li><math>p</math> is an extreme point of <math>K.</math></li>
<li><math>K \setminus \{p\}</math> is convex.</li>
<li><math>p</math> is not the midpoint of a non-degenerate line segment contained in <math>K.</math></li>
<li>for any <math>x, y \in K,</math> if <math>p \in [x, y]</math> then <math>x = p \text{ or } y = p.</math></li>
<li>if <math>x \in X</math> is such that both <math>p + x</math> and <math>p - x</math> belong to <math>K,</math> then <math>x = 0.</math></li>
<li><math>\{p\}</math> is a face of <math>K.</math></li>
</ol>
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