Minkowski sum of semi-convex domains in ${\Bbb R}^2$
Volume 411 / 2002
Abstract
\noindent The Minkowski sum of two sets $A$, $B$ in ${\sym R}^n$ is defined to be the set of all points of the form $a + b$ for $a \in A$ and $b \in B$. Due to its fundamental nature, the Minkowski sum is an important object in many practical application areas such as image processing, geometric design, robotics, etc. However, compared to the simplicity of the definition, a Minkowski sum of plane domains can have quite complicated topological and geometric features in general. This is the case even when the summands are relatively simple. For example, even if the summands are homeomorphic to the unit disk, their Minkowski sum need not be. We first introduce natural curve classes called {\it Minkowski classes}, and show that the set of all planar domains, called ${\cal M}$-domains, whose boundaries consist of a finite number of curves in a Minkowski class ${\cal M}$, is closed under Minkowski sum. Then we introduce the notion of {\it semi-convexity}\/ for plane domains, which extends convexity, and show that the Minkowski sum of semi-convex ${\cal M}$-domains is homeomorphic to the unit disk for any Minkowski class ${\cal M}$. We also show that, in some sense, the semi-convexity is the weakest condition ensuring that the Minkowski sum is homeomorphic to the unit disk. It is also shown that the set of all semi-convex ${\cal M}$-domains is closed under Minkowski sum for any Minkowski class ${\cal M}$. These results reveal a new topological behavior of Minkowski sum.
