On $C^1$-Regularity of Functions that Define $G$-Closure

In this paper we show that the functions which are used in the characterization of the G-closure or the Gθ-closure of sets of matrices are continuously differentiable. These regularity results are based on the observation by Ball, Kirchheim and Kristensen [1] that separate convexity and upper semidifferentiability imply continuous differentiability.