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Barrier cone

From Wikipedia, the free encyclopedia

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

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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

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The function

defined for each continuous linear functional 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 for which σK() is finite.

The set of continuous linear functionals for which σK() ≤ 1 is known as the polar set of K. The set of continuous linear functionals for which σK() ≤ 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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  • Aubin, Jean-Pierre; Frankowska, Hélène (2009). Set-Valued Analysis (Reprint of the 1990 ed.). Boston, MA: Birkhäuser Boston Inc. pp. xx+461. ISBN 978-0-8176-4847-3. MR 2458436.