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On Variations of the Convex Hull Operator

Given a collection ${\cal F}$ of subsets of $R^n$, consider the operator $\mbox{hull}_{{\cal F}}$ associating to a set $X \subset R^n$ the intersection of all elements of ${\cal F}$ containing $X$. The aim of this note is the study of the operator $\mbox{hull}_{{\cal F}}$ and especially its relationship with the {\em convex hull} operator in the special case when ${\cal F}$ is the set of all half-spaces of $R^n$.

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