Canonical semicontinuity-preserving extension for the Kuratowski embedding

Determine whether there exists a canonical extension associated with the Kuratowski embedding X→ℓ∞ that preserves lower semicontinuity of the extended function, analogous to the convex extension to the semihull sh(X) used in Section 2.2 to obtain lower semicontinuity of line integrals along curve fragments. Specifically, ascertain whether a well-defined canonical extension operator for functions on X to an appropriate geometric hull in ℓ∞ under the Kuratowski embedding can be constructed so that the lower semicontinuity properties required in the line-integral framework are satisfied.

Background

The paper develops a fragment-wise theory using Lipschitz free space F(X) and its semihull sh(X) to define canonical extensions of functions that preserve lower semicontinuity, which is crucial for establishing semicontinuity of line integrals over curve fragments. This machinery underpins key approximation and representation results.

In contrast, the commonly used Kuratowski embedding X→ℓ∞ does not directly support the same canonical convex extension scheme. The authors explicitly note that the natural convex extension is not well-defined in ℓ∞ and raise the question of whether any canonical extension method exists in that setting that preserves the necessary lower semicontinuity properties.