, Volume 58, Issue 1-2, pp 135-142

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

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

Received: September 22, 1998; revised version: December 22, 1998.