Barrier cone

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

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

  • 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.
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