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points of where a linear functional achieves its minimum on . Thus, if is a linear functional on and , then is an exposed face of .

An exposed point of is a point such that is an exposed face. That is, for all .

An exposed face is a face, but the converse is not true (see the figure). An exposed face of is convex if is convex.If is a face of , then is a face of if and only if is a face of .

Competing definitions

Some authors do not include and/or

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