The problem of optimal smoothing for convex functions
Author:
Mohammad Ghomi
Journal:
Proc. Amer. Math. Soc. 130 (2002), 2255-2259
MSC (2000):
Primary 26B25, 52A41
DOI:
https://doi.org/10.1090/S0002-9939-02-06743-6
Published electronically:
March 25, 2002
MathSciNet review:
1896406
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Abstract | References | Similar Articles | Additional Information
Abstract: A procedure is described for smoothing a convex function which not only preserves its convexity, but also, under suitable conditions, leaves the function unchanged over nearly all the regions where it is already smooth. The method is based on a convolution followed by a gluing. Controlling the Hessian of the resulting function is the key to this process, and it is shown that it can be done successfully provided that the original function is strictly convex over the boundary of the smooth regions.
- [Ev] Evans, L. Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. MR 99e:35001
- [Gh] Ghomi, M. Strictly Convex Submanifolds and Hypersurfaces of Positive Curvature, J. Differential Geom. 57 (2001), 239-271.
- [Gr] Gruber, P. Aspects of approximation of convex bodies. Handbook of convex geometry, Vol. A, B, 319-345, North-Holland, Amsterdam, 1993. MR 95b:52003
- [GT] Gilbarg, D.; Trudinger, N. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin-New York, 1983. MR 86c:35035
- [GW]
Greene, R.; Wu, H.
approximations of convex, subharmonic, and plurisubharmonic functions. Ann. Sci. EEcole Norm. Sup. (4) 12 (1979), no. 1, 47-84. MR 80m:53055
- [He] Helms, L. Brownian motion in a closed convex polygon with normal reflection. Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), no. 2, 199-209. MR 94d:60111
- [Hi] Hirsch, M. Differential topology. Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976. MR 56:6669
- [Ro] Rockafellar, R.Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970 MR 43:445
- [RV] Roberts, A.; Varberg, E. Convex functions. Pure and Applied Mathematics, Vol. 57. Academic Press, New York-London, 1973. MR 56:1201
- [Sc] Schneider, R. Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993. MR 94d:52007
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Additional Information
Mohammad Ghomi
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email:
ghomi@math.sc.edu
DOI:
https://doi.org/10.1090/S0002-9939-02-06743-6
Keywords:
Convex function,
convolution,
smooth approximation,
mollifier
Received by editor(s):
December 19, 1999
Published electronically:
March 25, 2002
Communicated by:
Bennett Chow
Article copyright:
© Copyright 2002
American Mathematical Society