Intersecting unions of convex sets in $R^{n}$

Abstract:

Let $\mathcal {C} = \{ {C_\alpha }:\alpha$ in some index set $I\}$ be a collection of convex sets, and let $\mathfrak {M} = \{ {C_\alpha } \cup {C_\beta }:\alpha \ne \beta ,{C_\alpha },{C_\beta }$ in $\mathcal {C}\}$. In this paper, various decomposition theorems are obtained for the set $\cap \mathfrak {M}$.