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
b(K):=\left\{\ell\inX\ast\left|\supx\langle\ell,x\rangle<+infty\right.\right\}.
The function
\sigmaK\colon\ell\mapsto\supx\langle\ell,x\rangle,
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.