Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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 $ \Delta_m=\{(t_0,\dots, t_m)\in \mathbf{R}^{m+1}: t_i\ge 0, \sum_{i=0}^mt_i=1\}$ be the standard $ m$-dimensional simplex and let $ \varnothing\ne S\subset \bigcup_{m=1}^\infty\Delta_m$. Then a function $ h\colon C\to \mathbf{R}$ with domain a convex set in a real vector space is $ S$-almost convex iff for all $ (t_0,\dots, t_m)\in S$ and $ x_0,\dots, x_m\in C$ the inequality

$\displaystyle h(t_0x_0+\dots+t_mx_m)\le 1+ t_0h(x_0)+\cdots+t_mh(x_m) $

holds. A detailed study of the properties of $ S$-almost convex functions is made. If $ S$ contains at least one point that is not a vertex, then an extremal $ S$-almost convex function $ E_S\colon \Delta_n\to \mathbf{R}$ is constructed with the properties that it vanishes on the vertices of $ \Delta_n$ and if $ h\colon \Delta_n\to \mathbf{R}$ is any bounded $ S$-almost convex function with $ h(e_k)\le 0$ on the vertices of $ \Delta_n$, then $ h(x)\le E_S(x)$ for all $ x\in \Delta_n$. In the special case $ S=\{(1/(m+1),\dotsc, 1/(m+1))\}$, the barycenter of $ \Delta_m$, very explicit formulas are given for $ E_S$ and $ \kappa_S(n)=\sup_{x\in\Delta_n}E_S(x)$. These are of interest, as $ E_S$ and $ \kappa_S(n)$ are extremal in various geometric and analytic inequalities and theorems.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26B25, 52A27, 39B72, 41A44, 51M16, 52A21, 52A40

Retrieve articles in all journals with MSC (2000): 26B25, 52A27, 39B72, 41A44, 51M16, 52A21, 52A40


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
PII: S 0002-9947(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