A simple closure condition for the normal cone intersection formula

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Necessary and sufficient conditions for stable conjugate duality (2006, background): “In recent years, it has been shown in [5, 6, 4] that Fenchel’s duality holds under a dual epigraph condition, which is strictly weaker than the usual interior-point conditions [1, 12], and the bounded linear regularity condition [2], which have played important roles in convex analysis and optimizat”
Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization (2008, background): “Note that (8) implies the well-known normal cone intersection formula (see [11]), NC∩g−1(−S)(x)=NC(x)+Ng−1(−S), ∀x ∈ C ∩ g−1(−S).”
An alternative formulation for a new closed cone constraint qualification (2006, background): “[5], [ 6 ], [11]) and prove that the results described above remain true.”