Find a concept
Start typing to search the mathematical index.
For convex sets with nonempty interior, algebraic and topological interiors coincide.
Let be a normed vector space and let be convex with nonempty interior. Then its algebraic interior equals its topological interior:
The inclusion is direct. For the reverse inclusion, translate so that and use convexity together with the interior-segment lemma.
Let be a vector space over (or ). A norm on is a function such that for all and scalars :
A pair is called a normed vector space.
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
Lemma. Let be a normed vector space and let be a convex set with nonempty interior. If and , then
where is the half-open segment from to .