Smoothness of certain metric projections on Hilbert space

Holmes, Richard B.

doi:10.1090/S0002-9947-1973-0326252-2

Smoothness of certain metric projections on Hilbert space
HTML articles powered by AMS MathViewer

by Richard B. Holmes PDF

Trans. Amer. Math. Soc. 184 (1973), 87-100 Request permission

Abstract:

A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hilbert space. The former mapping is also considered within the context of more general Banach spaces.

References

Edgar Asplund, Čebyšev sets in Hilbert space, Trans. Amer. Math. Soc. 144 (1969), 235–240. MR 253023, DOI 10.1090/S0002-9947-1969-0253023-7
E. Asplund and R. T. Rockafellar, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467. MR 240621, DOI 10.1090/S0002-9947-1969-0240621-X
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Richard B. Holmes, A course on optimization and best approximation, Lecture Notes in Mathematics, Vol. 257, Springer-Verlag, Berlin-New York, 1972. MR 0420367, DOI 10.1007/BFb0059450
J. B. Kruskal, Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc. 23 (1969), 697–703. MR 259752, DOI 10.1090/S0002-9939-1969-0259752-9
Jean-Jacques Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273–299 (French). MR 201952, DOI 10.24033/bsmf.1625
R. Tyrrell Rockafellar, Convex functions, monotone operators and variational inequalities, Theory and Applications of Monotone Operators (Proc. NATO Advanced Study Inst., Venice, 1968) Edizioni “Oderisi”, Gubbio, 1969, pp. 35–65. MR 0261415
M. M. Vaĭnberg, Some questions of the differential calculus in linear spaces, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 4(50), 55–102 (Russian). MR 0050800
Eduardo H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 237–341. MR 0388177