In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space.
It is closely related to the notions of support functions and polar sets.
Let X be a Banach space and let K be a non-empty subset of X.
The barrier cone of K is the subset b(K) of X∗, the continuous dual space of X, defined by The function defined for each continuous linear functional ℓ on X, is known as the support function of the set K; thus, the barrier cone of K is precisely the set of continuous linear functionals ℓ for which σK(ℓ) is finite.
The set of continuous linear functionals ℓ for which σK(ℓ) ≤ 1 is known as the polar set of K. The set of continuous linear functionals ℓ for which σK(ℓ) ≤ 0 is known as the (negative) polar cone of K. Clearly, both the polar set and the negative polar cone are subsets of the barrier cone.