On generalized $f$-projection and generalized $f$-prox-regularity on Banach spaces

This paper investigates the connection between the generalized projection ${\pi }_S$ and the generalized $f$-projection ${\pi }_S^f$. By introducing an $f$-proximal normal cone, defined via the generalized $f$-projection, we derive key properties of this set, generalizing established results on the classical $V$-proximal normal cone to Banach spaces. Additionally, we propose and examine a new regularity condition called uniform $f$-prox-regularity, which extends the standard notion of prox-regularity by leveraging the adaptability of $f$-projections.