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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.

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