Separating Hyperplanes of Convex Sets

Combining existing approaches, we provide a uniform way to describe all hyperplanes which separate (properly, or strong\-ly) a given pair of non\-emp\-ty convex sets $K_1$ and $K_2$ in the $n$-dimensional Euclidean space. The method is based on considering $(n\! -\! 1)$-dimensional subspaces which bound the set $K_1 \!-\! K_2$ and then using properties of the polar cone $(K_1 \!-\! K_2)^\circ$. First, we characterize all separating hyperplanes with given normal vectors, and then those containing a given point. We also describe the union of all hyperplanes separating (properly separating) a given pair of convex sets.