Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping

Kruskal, J. B.

doi:10.1090/S0002-9939-1969-0259752-9

Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping
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by J. B. Kruskal

Proc. Amer. Math. Soc. 23 (1969), 697-703

DOI: https://doi.org/10.1090/S0002-9939-1969-0259752-9

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References

Miriam Ayer, H. D. Brunk, G. M. Ewing, W. T. Reid, and Edward Silverman, An empirical distribution function for sampling with incomplete information, Ann. Math. Statist. 26 (1955), 641–647. MR 73895, DOI 10.1214/aoms/1177728423
D. E. Barton and C. L. Mallows, The randomization bases of the problem of the amalgamation of weighted means, J. Roy. Statist. Soc. Ser. B 23 (1961), 423–433. MR 132567
D. J. Bartholomew, A test of homogeneity for ordered alternatives, Biometrika 46 (1959), no. 1-2, 36–48. MR 104312, DOI 10.1093/biomet/46.1-2.36
D. J. Bartholomew, A test of homogeneity of means under restricted alternatives, J. Roy. Statist. Soc. Ser. B 23 (1961), 239–281. MR 143288
H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. MR 0124813
J. B. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis, Psychometrika 29 (1964), 1–27. MR 169712, DOI 10.1007/BF02289565
R. E. Miles, The complete amalgamation into blocks, by weighted means, of a finite set of real numbers, Biometrika 46 (1959), 317–327. MR 112167, DOI 10.1093/biomet/46.3-4.317
J. J. Moreau, Convexity and duality, Functional Analysis and Optimization, Academic Press, New York, 1966, pp. 145–169. MR 0217617
M. Z. Nashed, A decomposition relative to convex sets, Proc. Amer. Math. Soc. 19 (1968), 782–786. MR 234268, DOI 10.1090/S0002-9939-1968-0234268-3
R. R. Phelps, Convex sets and nearest points, Proc. Amer. Math. Soc. 8 (1957), 790–797. MR 87897, DOI 10.1090/S0002-9939-1957-0087897-7
R. R. Phelps, Convex sets and nearest points. II, Proc. Amer. Math. Soc. 9 (1958), 867–873. MR 104139, DOI 10.1090/S0002-9939-1958-0104139-5
Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Constance van Eeden, Maximum likelihood estimation of partially or completely ordered parameters. I, Indag. Math. 19 (1957), 128–136. Nederl. Akad. Wetensch. Proc. Ser. A 60. MR 0083869
Constance van Eeden, Note on two methods for estimating ordered parameters of probability distributions, Indag. Math. 19 (1957), 506–512. Nederl. Akad. Wetensch. Proc. Ser. A 60. MR 0102872
Hans S. Witsenhausen, A minimax control problem for sampled linear systems, IEEE Trans. Automatic Control AC-13 (1968), 5–21. MR 0230562, DOI 10.1109/tac.1968.1098788