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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
Published electronically: March 25, 2002
MathSciNet review: 1896406
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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.

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  • [Ev] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • [Gh] Ghomi, M. Strictly Convex Submanifolds and Hypersurfaces of Positive Curvature, J. Differential Geom. 57 (2001), 239-271.
  • [Gr] Peter M. Gruber, Aspects of approximation of convex bodies, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 319–345. MR 1242984
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [GW] R. E. Greene and H. Wu, 𝐶^{∞} approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376
  • [He] L. L. Helms, 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 1190319,
  • [Hi] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
  • [Ro] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
  • [RV] A. Wayne Roberts and Dale E. Varberg, Convex functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 57. MR 0442824
  • [Sc] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521

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Additional Information

Mohammad Ghomi
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

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