Almost Regular Metrics on Groups and Lipschitz-Continuity of Distance Transforms

In this article, the almost inner (resp. lower, upper) regularity of metrics on a group is proposed. In terms of these regularities, a sufficient and necessary condition for distance transforms to be Lipschitz-1 continuous is given, and relations between balls with different centres and radii are discussed. It turns out that the three almost regularities of metrics introduced and the results obtained here might provide useful tools in discrete analysis, mathematical morphology and image analysis, etc.