Abstract
In this paper we shall investigate the positive center set \({\mathfrak {P}}(\gamma )\) of a convex curve \(\gamma \) and show that \({\mathfrak {P}}(\gamma )\) has only one point if and only if \(\gamma \) is a circle; \({\mathfrak {P}}(\gamma )\) is a segment if and only if \(\gamma \) is a sausage curve; if \(\gamma \) is a strictly convex non-circular curve, then \({\mathfrak {P}}(\gamma )\) is a domain of positive area; and furthermore, if \(\gamma \) is a constant width curve, then \({\mathfrak {P}}(\gamma )\) is its inner parallel body \(K_{-r_1}\).
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Acknowledgments
This work was supported by the National Science Foundation of China (No. 11171254) and a Grant of “The First-class Discipline of Universities in Shanghai.” The authors would like to thank the referees for their careful reading of the original manuscript of this paper and giving them some helpful suggestions and invaluable comments.
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Huang, P., Pan, S. & Yang, Y. Positive Center Sets of Convex Curves. Discrete Comput Geom 54, 728–740 (2015). https://doi.org/10.1007/s00454-015-9715-9
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DOI: https://doi.org/10.1007/s00454-015-9715-9