Barrier cone

Definition

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 \in X^{\ast} ,\left|, \sup_{x \in K} \langle \ell, x \rangle

Related notions

The function

:\sigma_{K} \colon \ell \mapsto \sup_{x \in K} \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 *σ*K(ℓ) is finite.

The set of continuous linear functionals ℓ for which *σ*K(ℓ) ≤ 1 is known as the polar set of K. The set of continuous linear functionals ℓ for which *σ*K(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.

References

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