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Counterexamples to Smoothing Convex Functions

Published online by Cambridge University Press:  20 November 2018

Patrick Adrian Neale Smith
Patrick Adrian Neale Smith*
Affiliation:
University of Toronto, Toronto Ont. M5S 1A1
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Abstract

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Greene and Wu have shown that any continuous strongly convex function on a Riemannian manifold can be uniformly approximated by infinitely differentiable strongly convex functions. This result is not true if the word “strongly” is omitted; in this paper, we give examples of manifolds on which convex functions cannot be approximated by convex functions (k = 0, 1,2,...).

Keywords

53C99

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 3 , 01 September 1986, pp. 308 - 313
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Fornaess, J. E., Regularizations of Plurisubharmonic Functions, Mathematische Annalen, 259 (1982), pp. 119–123.Google Scholar
2. Greene, R. E. and Wu, H., Convex functions and Manifolds of Positive Curvature, Acta Mathematica, 137(1976), pp. 209–245.Google Scholar
3. Greene, R. E. and Wu, H., Approximations of Convex, Subharmonic, and Plurisubharmonic Functions, Ann. scient. Ec. Norm. Sup. (4). 12 (1979), pp. 47–84.Google Scholar
4. Greene, R. E. and Wu, H., Gap Theorems for Noncompact Riemannian Manifolds, Duke Math. J., 49 (1982), pp. 731–756.Google Scholar
5. Greene, R. E. and Wu, H., On the subharmonic ity and plurisubharmonicity of geodesic ally convex functions, Indiana Univ. Math. J., 22 (1973), pp. 641–653.Google Scholar
6. Greene, R. E. and Wu, H., Integrals of Subharmonic Functions on Manifolds of Nonnegative Curvature, Inven. Math., 27 (1974), pp. 265–298.Google Scholar