Let $\{f_n\}$ be a sequence of uniformly upper bounded functions defined on $\mathbb{R}$ s.t. for every $x\in\mathbb{R}$

$$f_n(x)\to f(x),~ n\to\infty$$

Define $g_n$ and $g$ as the concave envelope of $f_n$ and $f$, i.e. $g_n$ ($g$) is defined as the smallest concave function which dominates $f_n$ ($f$).

Then my question is whether we have for every $x\in\mathbb{R}$

$$g_n(x)\to g(x)$$

If not, could the uniform convergence of $f_n$ to $f$ imply $g_n(x)\to g(x)$? Many thanks!

  • $\begingroup$ The answer is yes, uniform convergence is not needed, but you have to be careful with the points where the concave envelope tends to $-\infty$. $\endgroup$ – Alexandre Eremenko May 10 '14 at 20:01
  • 1
    $\begingroup$ Thx a lot for your reply, could you give the related reference for this proof or some details? $\endgroup$ – CodeGolf May 10 '14 at 22:01

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