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Aequationes mathematicae

, Volume 58, Issue 1–2, pp 135–142 | Cite as

The local stability of convexity, affinity and of the Jensen equation

  • M. Laczkovich
Article

Summary.

Let \( C_D, A_D, J_D \) denote the smallest constants involved in the stability of convexity, affinity and of the Jensen equation of functions defined on a convex subset D of \( {\Bbb R}^n \). By a theorem of J. W. Green, \( C_D \le c\cdot \log (n+1) \) for every convex \( D\subset {\Bbb R}^n \), where c is an absolute constant. We prove that the lower estimate \( C_D \ge c\cdot \log (n+1) \) is also true, supposing that int \( D \neq {\not 0} \).¶We show that \( A_D \le 2 C_D \) and \( A_D \le J_D \le 2A_D \) for every convex \( D\subset {\Bbb R}^n \). The constant \( J_D \) is not always of the same order of magnitude as \( C_D \); for example \( J_D = 1 \) if \( D ={\Bbb R}^n \). We prove that there are convex sets (e.g. the n-dimensional simplex) with \( J_D \ge c\cdot \log n \).

Keywords

Convex Subset Lower Estimate Local Stability Absolute Constant Small Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel, 1999

Authors and Affiliations

  • M. Laczkovich
    • 1
  1. 1.Department of Analysis, Eötvös Loránd University, Rákóczi út 5, H-1088 Budapest, Hungary, e-mail: laczk@cs.elte.huHungary

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