Characterizations of strictly monotone sets
HTML articles powered by AMS MathViewer
- by Jean Chan Stanek PDF
- Proc. Amer. Math. Soc. 64 (1977), 112-117 Request permission
Abstract:
One natural generalization of convex sets is the concept of strictly monotone sets. In this paper, Tietze’s characterization theorem on convex sets is generalized to strictly monotone sets. The main result is that a closed connected set in the Euclidean plane ${E_2}$ is strictly monotone if and only if it is locally strictly monotone. This theorem is shown to hold in ${E_n}$ with additional hypotheses.References
- V. L. Klee Jr., Convex sets in linear spaces, Duke Math. J. 18 (1951), 443–466. MR 44014 J. Slaninger, Subconvex sets, Dissertation, Univ. of California, Los Angeles, 1970.
- Jean Chan Stanek, Tietze-type theorems on monotone increasing sets, Proc. Amer. Math. Soc. 54 (1976), 286–290. MR 388244, DOI 10.1090/S0002-9939-1976-0388244-5
- Heinrich Tietze, Über Konvexheit im kleinen und im großen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928), no. 1, 697–707 (German). MR 1544985, DOI 10.1007/BF01181191
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 64 (1977), 112-117
- MSC: Primary 52A20; Secondary 52A10
- DOI: https://doi.org/10.1090/S0002-9939-1977-0470858-6
- MathSciNet review: 0470858