Topologies on Closed and Closed Convex Sets pp 270-305 | Cite as

# The Slice Topology for Convex Functions

## Abstract

Consistent with the presentation of Chapter 7, slice convergence of a net of lower semicontinuous proper convex functions defined on a normed linear space X means the convergence of their epigraphs with respect to the slice topology on C(*X* × *R*) as determined by the box norm on *X* × *R*. Similarly, we may speak of dual slice convergence of functions in *Γ**(*X**). A major result of this chapter shows that slice convergence in *Γ*(*X*) implies and is implied by dual slice convergence in *Γ**(*X**). From this result, and our presentation of the slice and dual slice topologies as weak topologies in Chapter 2, Wijsman convergence of a net of closed convex sets with respect to all norms equivalent to a given initial norm implies Wijsman convergence of the polars with respect to the dual norms. But most importantly, slice convergence is intrinsic to convex duality, and we present a number of characterizations of the topology in terms of the lower semicontinuity of natural geometrically defined multifunctions on *Γ*(*X*) and on C(*X*).

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