# 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.

## 9 Citations

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## References

SHOWING 1-10 OF 39 REFERENCES

Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings

- Mathematics
- 1988

Concepts of lower semicontinuity and continuous selections for convex valued multifunctions

- Mathematics
- 1992

A characterization of reflexive spaces by means of continuous approximate selections for metric projections

- Mathematics
- 1989

Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections

- Mathematics
- 1983

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)

- Mathematics
- 1988

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

- Mathematics
- 1991

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

- Mathematics
- 1997

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

- Mathematics
- 1995

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…