Lower semicontinuity and generic behavior of δ_full

Ascertain whether the microstates free entropy dimension for full types δ_full enjoys a lower semi-continuity property (for example, with respect to the Wasserstein topology on S_{m,R}(T_ω)), and characterize its generic behavior for types supported in the operator-norm ball (D_R)^m.

Background

The paper establishes several topological properties: χ_full is weak-* upper semicontinuous and Ent_full is lower semicontinuous with respect to Wasserstein distance. It also shows genericity statements for Ent_full and χ_full.

In contrast, δ_full is known not to be upper semicontinuous in general, but whether it has any lower semicontinuity remains unknown. Understanding its typical (generic) value would clarify how dimension-like invariants behave on the type space endowed with the Wasserstein metric.