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 \( J_D \ge c\cdot \log n \).

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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