Support cones and convexity of sets in ${\mathbb {R}}^n$
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- by Robert Huotari and Junning Shi
- Proc. Amer. Math. Soc. 124 (1996), 2405-2414
- DOI: https://doi.org/10.1090/S0002-9939-96-03347-3
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Abstract:
We discuss several metric characterizations of convexity of sets in non-smooth finite-dimensional Banach spaces. We describe a setting in which convexity is equivalent to the rotation-invariance of various properties, including almost convexity, radial continuity of the metric projection, and Chebyshevity. One of the tools used is a generalization of norm-smoothness which involves support cones of the unit ball.References
- Dan Amir and Frank Deutsch, Suns, moons, and quasi-polyhedra, J. Approximation Theory 6 (1972), 176–201. MR 364982, DOI 10.1016/0021-9045(72)90073-1
- J. Blatter, P. D. Morris, and D. E. Wulbert, Continuity of the set-valued metric projection, Math. Ann. 178 (1968), 12–24. MR 228984, DOI 10.1007/BF01350621
- Dietrich Braess, Nonlinear approximation theory, Springer Series in Computational Mathematics, vol. 7, Springer-Verlag, Berlin, 1986. MR 866667, DOI 10.1007/978-3-642-61609-9
- Bruno Brosowski and Frank Deutsch, Some new continuity concepts for metric projections, Bull. Amer. Math. Soc. 78 (1972), 974–978. MR 304948, DOI 10.1090/S0002-9904-1972-13073-8
- Bruno Brosowski and Frank Deutsch, On some geometric properties of suns, J. Approximation Theory 10 (1974), 245–267. MR 358180, DOI 10.1016/0021-9045(74)90122-1
- Bruno Brosowski and Frank Deutsch, Radial continuity of set-valued metric projections, J. Approximation Theory 11 (1974), 236–253. MR 350285, DOI 10.1016/0021-9045(74)90016-1
- F. Deutsch, The Geometry of Banach Spaces and its Application to Approximation Theory, 1979.
- Frank Deutsch, The convexity of Chebyshev sets in Hilbert space, Topics in polynomials of one and several variables and their applications, World Sci. Publ., River Edge, NJ, 1993, pp. 143–150. MR 1276957
- Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094, DOI 10.1007/BFb0082079
- John R. Giles, Convex analysis with application in the differentiation of convex functions, Research Notes in Mathematics, vol. 58, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 650456
- Robert Huotari and Wu Li, Continuity of metric projection, Pólya algorithm, strict best approximation, and tubularity of convex sets, J. Math. Anal. Appl. 182 (1994), no. 3, 836–856. MR 1272156, DOI 10.1006/jmaa.1994.1124
- R. Huotari and W. Li, Continuity of metric projection and geometric consequences, (submitted)
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Victor Klee, Convexity of Chevyshev sets, Math. Ann. 142 (1960/61), 292–304. MR 121633, DOI 10.1007/BF01353420
- L. P. Vlasov, Chebyshev sets in Banach spaces, Dokl. Akad. Nauk SSSR 141 (1961), 19–20 (Russian). MR 0131748
- L. P. Vlasov, On almost convex sets in Banach spaces, Dokl. Akad. Nauk SSSR 163 (1965), 18–21 (Russian). MR 0206679
- L. P. Vlasov, Čebyšev sets and approximately convex sets, Mat. Zametki 2 (1967), 191–200 (Russian). MR 215060
- L. P. Vlasov, On Čebyšev sets, Dokl. Akad. Nauk SSSR 173 (1967), 491–494 (Russian). MR 0215059
- L. P. Vlasov, Almost convex and Čebyšev sets, Mat. Zametki 8 (1970), 545–550 (Russian). MR 276736
Bibliographic Information
- Robert Huotari
- Affiliation: Department of Mathematics, Idaho State University, Pocatello, Idaho 83209
- Address at time of publication: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900
- Email: huotari@isu.edu
- Junning Shi
- Affiliation: Permanent address : Allianz Insurance Company, 3400 Riverside Dr., Suite 300, Burbank, California 91505
- Received by editor(s): April 20, 1994
- Received by editor(s) in revised form: February 10, 1995
- Communicated by: J. Marshall Ash
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2405-2414
- MSC (1991): Primary 41A65; Secondary 41A62
- DOI: https://doi.org/10.1090/S0002-9939-96-03347-3
- MathSciNet review: 1327019