Almost Regular Metrics on Groups and Lipschitz-Continuity of Distance Transforms
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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.
KeywordsDistance transform Lipschitz-continuity Regularity of metrics Discrete analysis
Mathematics Subject Classification46B99 68R99 65D18 22A20
The authors would like to express sincere thanks to the referees for their careful reading of the original and the revised manuscripts, their valuable suggestions and recommendations on the content and the references, and also for their pointing out errors and typos, which improve the article. The study is supported by the National Natural Science Foundation of China (Nos. 11671293, 11271282).
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