Variational Problems in the Class of Domains under Normal Geometric Property
Mohammed Barkatou · HAL (Le Centre pour la Communication Scientifique Directe) · 2026
This work explores a family of geometric variational problems posed in the class O C of domains satisfying the geometric normal property with respect to a fixed convex set C. Building on the radial structure offered by this class, we study in detail a Cheeger-type problem with a constraint on the largest included convex set, as well as several natural variants. We establish existence, uniqueness, and geometric characterization results for minimizers. Furthermore, we introduce and analyze a natural reciprocal mapping from ∂Ω \ C to ∂C that serves as the geometric inverse of the radial parametrization. This mapping allows us to define a dual thickness function δ : ∂Ω \ C → R + which measures the thickness of the domain from its outer boundary. This new notion provides novel insights into the structure of domains in O C , yields dual formulations of variational problems, and opens several research perspectives that we identify throughout the work.