  For convex sets $K$ and $L$ in ${\mathbb{R}}^d$ we define $R_L(K)$ to be the
convex hull of all points belonging to $K$ but not to the interior of $L$.
Cutting-plane methods from integer and mixed-integer optimization can be
expressed in geometric terms using functionals $R_L$ with appropriately chosen
sets $L$. We describe the geometric properties of $R_L(K)$ and characterize
those $L$ for which $R_L$ maps polyhedra to polyhedra. For certain natural
classes ${\mathcal{L}}$ of convex sets in ${\mathbb{R}}^d$ we consider the
functional $R_{\mathcal{L}}$ given by $R_{\mathcal{L}}(K):= \bigcap_{L \in
{\mathcal{L}}}R_L(K)$. The functional $R_{\mathcal{L}}$ can be used to define
various types of closure operations considered in the theory of cutting planes
(such as the Chv\'atal closure, the split closure as well as generalized split
closures recently introduced by Andersen, Louveaux and Weismantel). We study
conditions on ${\mathcal{L}}$ under which $R_{\mathcal{L}}$ maps rational
polyhedra to rational polyhedra. We also describe the limit of the sequence of
sets obtained by iterative application of $R_{\mathcal{L}}$ to $K$. A part of
the presented material gives generalized formulations and unified proofs of
several recent results obtained by various authors.

Optimization and Control;Metric Geometry