A general theory of almost convex functions
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- by S. J. Dilworth, Ralph Howard and James W. Roberts PDF
- Trans. Amer. Math. Soc. 358 (2006), 3413-3445 Request permission
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 \[ h(t_0x_0+\dots +t_mx_m)\le 1+ t_0h(x_0)+\cdots +t_mh(x_m) \] 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
- Piotr W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76–86. MR 758860, DOI 10.1007/BF02192660
- 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, DOI 10.1006/aima.1999.1836
- 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, DOI 10.1006/aima.2001.2058
- Donald H. Hyers, George Isac, and Themistocles M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, vol. 34, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1639801, DOI 10.1007/978-1-4612-1790-9
- D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821–828. MR 49962, DOI 10.1090/S0002-9939-1952-0049962-5
Additional Information
- S. J. Dilworth
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 58105
- Email: dilworth@math.sc.edu
- Ralph Howard
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 88825
- Email: howard@.sc.edu
- James W. Roberts
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- Email: roberts@math.sc.edu
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 3413-3445
- MSC (2000): Primary 26B25, 52A27; Secondary 39B72, 41A44, 51M16, 52A21, 52A40
- DOI: https://doi.org/10.1090/S0002-9947-06-04061-X
- MathSciNet review: 2218982