Exposed point

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

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

There are two weaker notions, that of extreme point and that of support point of C {\displaystyle C}.

See also

• Exposed face