Differentiation of sets - The general case

In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limit theorems were established for local empirical processes near the boundary of compact convex sets $K$ in $\mathbb{R}$. The limit processes were shown to live on the normal cylinder $\mathrm{\Sigma }$ of $K$, respectively on a class of set-valued derivatives in $\mathrm{\Sigma }$. The latter result was based on the concept of differentiation of sets at the boundary $\mathrm{\partial }K$ of $K$, which was developed in Khmaladze (2007). Here, we extend the theory of set-valued derivatives to boundaries $\mathrm{\partial }F$ of rather general closed sets $F\subset \R$, making use of a local Steiner formula for closed sets, established in Hug, Last and Weil (2004).