New Properties for Full Convex Sets and Full Convex Hulls

Full convexity has been recently proposed as an alternative definition of digital convexity in $Z^n$. In contrast to classical definitions, fully convex sets are always connected and even simply connected whatever the dimension, while remaining digitally convex in the usual sense. In order to better understand the properties of full convexity, we present here two new and radically different characterizations of full convexity. The first one mimicks the usual continuous convexity via segments inclusion. We show an equivalence of full convexity with this segment convexity in dimensions 1 and 2, and counterexamples starting from dimension 3. If we now ask that the cells touched by all d-simplices (instead of just 2-simplices aka segments) are within the cells touched by the digital set, we achieve an equivalence in arbitrary dimension d. The second characterization is recursive with respect to the dimension, and relies on convexity of its axis-aligned projections. We provide several applications of these characterizations: the full convexity of balls and subsets of the hypercube, a natural measure of full convexity for digital sets, a new and faster algorithm to check the full convexity of digital sets. Finally, we study the main drawback of full convexity: its envelope operator may not be an increasing operator. We characterize fully convex sets that have anti-monotonous subsets, and we show that they must be thin in a precised sense..

J. Math. Imaging Vis., 67(1): 8, 2025.