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Barrier cone

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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.

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 ) := { ℓ ∈ X ∗ | sup x ∈ K ⟨ ℓ , x ⟩ < + ∞ } . {\displaystyle b(K):=\left\{\ell \in X^{\ast }\,\left|\,\sup _{x\in K}\langle \ell ,x\rangle <+\infty \right.\right\}.}

Related notions

The function

σ K : ℓ ↦ sup x ∈ K ⟨ ℓ , x ⟩ , {\displaystyle \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 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.

References

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