Exposed point

In mathematics, an exposed point of a convex set $$C$$ is a point $$x\in C$$ at which some continuous linear functional attains its strict maximum over $$C$$. Such a functional is then said to expose $$x$$. There can be many exposing functionals for $$x$$. The set of exposed points of $$C$$ is usually denoted $$\exp(C)$$.

A stronger notion is that of strongly exposed point of $$C$$ which is an exposed point $$x \in C$$ such that some exposing functional $$f$$ of $$x$$ attains its strong maximum over $$C$$ at $$x$$, i.e. for each sequence $$(x_n) \subset C$$ we have the following implication: $$f(x_n) \to \max f(C) \Longrightarrow \|x_n -x\| \to 0$$. The set of all strongly exposed points of $$C$$ is usually denoted $$\operatorname{str}\exp(C)$$.

There are two weaker notions, that of extreme point and that of support point of $$C$$.