(Solution Download) If S is a convex set in a normed linear

If S is a convex set in a normed linear space, ext (S) ? (S).
The converse is not true in general; not all boundary points are extreme points. However, boundary points and extreme points coincide when a set is strictly convex. A set S in a normed linear space is called strictly convex if the straight line between any two points lies in the interior of the set. More precisely, S is strictly convex if for every x, y in X with x?y,

Note that the interior of a convex set is always strictly convex (exercise 1.217). Therefore the additional requirement of strict convexity applies only to boundary points, implying that the straight line between any two boundary points lies in the interior of the set. Hence the boundary of a strictly convex set contains no line segment and every boundary point is an extreme point.

 

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