Lower Semicontinuity Concepts, Continuous Selections, and Set Valued Metric Projections

@article{Brown2002LowerSC,
  title={Lower Semicontinuity Concepts, Continuous Selections, and Set Valued Metric Projections},
  author={A. L. Brown and Frank Deutsch and V. Indumathi and Petar S. Kenderov},
  journal={J. Approx. Theory},
  year={2002},
  volume={115},
  pages={120-143}
}
A number of semicontinuity concepts and the relations between them are discussed. Characterizations are given for when the (set-valued) metric projection P"M onto a proximinal subspace M of a normed linear space X is approximate lower semicontinuous or 2-lower semicontinuous. A geometric characterization is given of those normed linear spaces X such that the metric projection onto every one-dimensional subspace has a continuous C"0(T) and L"1(@m) that have this property are determined. 
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References

SHOWING 1-10 OF 39 REFERENCES
Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections
Two new continuity properties for set-valued mappings are defined which are weaker than lower semicontinuity. One of these properties characterizes when approximate selections exist. A few selection
CHARACTERIZATION OF CONTINUOUS SELECTIONS FOR METRIC PROJECTIONS IN C(X)
In 1969, Lazar, Morris and Wulbert gave a necessary condition of OCC(X) whose metric projection P_G has s continuous selection. In this paper, we show that the necessary condition mentioned above is
Continuous Selections for Metric Projections and Interpolating Subspaces
Contents: This monograph deals with various intrinsic characterizations of those subspaces G of C o(T) whose metric projections P G have continuous selections. We have a systematic development of the
The Derived Mappings and the Order of a Set-Valued Mapping Between Topological Spaces
Previous investigations of, in particular, continuous selections have led to the definition of the derived mappings and, here, the order of a set-valued mapping between topological spaces. The
Metric Projections in Spaces of Integrable Functions
The paper is concerned with the calculation of the derived mapping of the metric projection onto a finite dimensional subspace of a space of integrable functions. Abstract results for quotient spaces
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