On the Cellular Convexity of Complexes

In this paper we discuss cellular convexity of complexes. A new definition of cellular convexity is given in terms of a geometric property. Then it is proven that a regular complex is celiularly convex if and only if there is a convex plane figure of which it is the cellular image. Hence, the definition of cellular convexity by Sklansky [7] is equivalent to the new definition for the case of regular complexes. The definition of Minsky and Papert [4] is shown to be equivalent to our definition. Therefore, aU definitions are virtually equivalent. It is shown that a regular complex is cellularly convex if and only if its minimum-perimeter polygon does not meet the boundary of the complex. A 0(n) time algorithm is presented to determine the cellular convexity of a complex when it resides in n × m cells and is represented by the run length code.