A General Result About Inner Regularization of Sets

We generalize a recent result of C. Nour, H. Saoud and J. Takche [Regularization via sets satisfying the interior sphere condition, J. Convex Analysis 25(1) (2018) 1–19], where the authors provided an inner approximation of a subset S of the n-dimensional real space $\mathbb{R}^n$ by sets satisfying the interior sphere condition. The main idea is to replace the convex cones used there by curved ones which leads us to weaken the assumptions and cover more sets S.