Topologies on Closed and Closed Convex Sets pp 235-269 | Cite as

# The Attouch-Wets Topology for Convex Functions

## Abstract

A proper lower semicontinuous convex function *f* on a normed linear space *X*, identified with its epigraph, is a nonempty closed convex subset of *X* × *R* containing no vertical lines that recedes in the vertical direction: whenever (*x*,α) ∈ epi *f*, and β > α, then (*x*,β) ∈ epi *f*. Hyperspace topologies defined on C(*X* × *R*) naturally induce topologies on *Γ*(*X*). In this chapter, we look in some depth at an important topology on *Γ*(*X*) that arises in this way: the Attouch-Wets topology. For one thing, the convergence of a sequence of closed convex sets corresponds to the convergence of the associated sequences of indicator functions, support functions, and distance functions. Furthermore, convergence of a sequence of proper lower semicontinuous convex functions in the Attouch-Wets topology implies and is implied by the Attouch-Wets convergence of functions dual to the originals in *Γ**(*X**). Continuity of polarity for convex sets easily falls out of this result for convex functions. Thus, the Attouch-Wets topology is stable with respect to duality. Moreover, the strength of Attouch-Wets topology, as well as its overall tractability in terms of estimation, make it a highly potent convergence concept applicable to convex optimization and approximation problems.

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