A general theory of almost convex functions

Authors:
S. J. Dilworth, Ralph Howard and James W. Roberts

Journal:
Trans. Amer. Math. Soc. **358** (2006), 3413-3445

MSC (2000):
Primary 26B25, 52A27; Secondary 39B72, 41A44, 51M16, 52A21, 52A40

Published electronically:
March 1, 2006

MathSciNet review:
2218982

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -** almost convex** iff for all and the inequality

**1.**Piotr W. Cholewa,*Remarks on the stability of functional equations*, Aequationes Math.**27**(1984), no. 1-2, 76–86. MR**758860**, 10.1007/BF02192660**2.**S. J. Dilworth, Ralph Howard, and James W. Roberts,*Extremal approximately convex functions and estimating the size of convex hulls*, Adv. Math.**148**(1999), no. 1, 1–43. MR**1736640**, 10.1006/aima.1999.1836**3.**S. J. Dilworth, Ralph Howard, and James W. Roberts,*Extremal approximately convex functions and the best constants in a theorem of Hyers and Ulam*, Adv. Math.**172**(2002), no. 1, 1–14. MR**1943899**, 10.1006/aima.2001.2058**4.**Donald H. Hyers, George Isac, and Themistocles M. Rassias,*Stability of functional equations in several variables*, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1639801****5.**D. H. Hyers and S. M. Ulam,*Approximately convex functions*, Proc. Amer. Math. Soc.**3**(1952), 821–828. MR**0049962**, 10.1090/S0002-9939-1952-0049962-5

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Additional Information

**S. J. Dilworth**

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

Email:
dilworth@math.sc.edu

**Ralph Howard**

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

Email:
howard@.sc.edu

**James W. Roberts**

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

Email:
roberts@math.sc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-06-04061-X

Keywords:
Convex hulls,
convex functions,
approximately convex functions,
normed spaces,
Hyers-Ulam Theorem

Received by editor(s):
January 31, 2001

Received by editor(s) in revised form:
May 13, 2004

Published electronically:
March 1, 2006

Additional Notes:
The research of the second author was supported in part by ONR Grant N00014-90-J-1343 and ARPA-DEPSCoR Grant DAA04-96-1-0326

Article copyright:
© Copyright 2006
American Mathematical Society