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The set lin(Ω) is convex whenever Ω is convex.
Let be a vector space and let be convex.
Proposition: The linear closure is convex.
Context: The definition of is built from line segments. Convexity of ensures that "segments staying in " is stable under convex combinations, which yields convexity of .
A vector space over a field is a set with operations (functions) and , and an element , satisfying the following for all and :
Let be a vector space (typically over ). A set is convex if for all and all we have
Context. Equivalently, is convex iff it contains every line segment joining any two of its points. Convexity is the core geometric notion underlying convex analysis and optimization.
Let be a vector space and let .
Let be a real vector space. A vector is a convex combination of points if there exist scalars with
such that
Context. Convex combinations describe the points obtained by repeatedly taking "weighted averages." They generate the convex hull of a set.