Properties of the metric projection on weakly Vial-convex sets and parametrization of set-valued mappings with weakly convex images. (English. Russian original) Zbl 1117.52001
Math. Notes 80, No. 4, 461-467 (2006); translation from Mat. Zametki 80, No. 4, 483-489 (2006).
Summary: We continue our study of the class of weakly convex sets (in the sense of Vial) which was stated in [G. E. Ivanov, Izv. Math. 69, No. 6, 1113–1135 (2005; Zbl 1104.52002)]. For points in a sufficiently small neighborhood of a closed weakly convex subset in Hilbert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.
MSC:
| 52A01 | Axiomatic and generalized convexity |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A27 | Approximation by convex sets |
| 93B15 | Realizations from input-output data |
| 47H04 | Set-valued operators |
Keywords:
weakly convex sets; metric projection; set-valued mapping; modulus of continuity; Chebyshev layer; Hausdorff distanceCitations:
Zbl 1104.52002References:
| [1] | G. E. Ivanov, ”Sets weakly convex in the sense of Vial and in the sense of Efimov and Stechkin,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 69 (2005), no. 6, 35–60. |
| [2] | J. W. Daniel, ”The continuity of metric projections as functions of the data,” J. Approximation Theory, 12 (1974), 234–239. · Zbl 0297.41021 · doi:10.1016/0021-9045(74)90065-3 |
| [3] | E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis [in Russian], Fizmatlit, Moscow, 2004. · Zbl 1181.26028 |
| [4] | A. D. Ioffe, ”Representation of set-valued mappings – II. Application to differential inclusions,” SIAM J. Control Optim., 21 (1983), 641–651. · Zbl 0539.49009 · doi:10.1137/0321038 |
| [5] | J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, 1984. · Zbl 0538.34007 |
| [6] | A. Ornelas, ”Parametrization of Caratheodory multifunctions,” Rend. Sem. Mat. Univ. Padova, 83 (1990), 34–43. · Zbl 0708.28005 |
| [7] | A. LeDonne and M. V. Marchi, ”Representation of Lipschitz compact convex valued mappings,” Rend. Acad. Naz. Lincei, 68 (1980), 278–280. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
