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}

.