The Variable Radius Form of the Extended Exterior Sphere Condition
Chadi Nour, Jean Takche
TL;DR
The paper extends the exterior sphere condition to a variable radius function $r(\cdot)$ and proves that if a nonempty closed set $A$ satisfies this extended exterior $r(\cdot)$-sphere condition, then its complement $A^c$ is the union of closed balls with a radius function $\rho(x)=\min\left\{ \frac{r(a)}{2} : a\in \operatorname{proj}_A(x) \right\}$. This generalizes the constant-radius result from prior work and shows lower semicontinuity of $\rho$, providing a structural decomposition of $A^c$ in terms of projections onto $A$. A second main contribution demonstrates that both constant- and variable-radius forms of the extended exterior sphere condition belong to the $S$-convexity regularity class, linking these geometric properties to proximal analysis and prox-regularity. The results unify and extend existing exterior sphere and union-of-balls frameworks, recover classical cases when $A$ is regular closed, and offer a cohesive regularity toolbox for finite-dimensional variational analysis.
Abstract
We introduce a variable radius form of the extended exterior sphere condition of [16], and then, we prove that the complement of a closed set satisfying this new property is nothing but the union of closed balls with lower semicontinous radius function. This generalizes, to the variable radius case, the main result of [16], namely, [16, Theorem 1.2]. On the other hand, as it is shown in [14,15] for prox-regularity, the exterior sphere condition, and the union of closed balls property, we prove that the constant and the variable radius forms of the extended exterior sphere condition belong to the S-convexity regularity class.