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Taking algebraic interior preserves convexity
Let be a vector space and let be convex. Then its algebraic interior
is convex. This includes the case in which the core is empty.
This is the algebraic analogue of the fact that the interior of a convex set in a normed space is convex; compare convexity of interiors and closures.
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 metric space and let .
The interior of , denoted or , is defined by
Let be a normed vector space and a convex set. Then: