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Can any convex function on a bounded interval be approximated uniformly by a sequence of strictly convex functions?

xyz
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    What do you mean by an approximation? How would you approximate $f=1_{{0}}$ on $[0,1]$? – copper.hat Apr 15 '20 at 00:17
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    @copper.hat: This function is already strictly convex (I assume you mean the function with $f(0) = 0$ and $f(x) = \infty$ for $x \ne 0$). – gerw Apr 16 '20 at 09:10
  • @gerw: I meant the standard characteristic function which is zero outside of the specified set. – copper.hat Apr 16 '20 at 17:33

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Take your existing sequence and add $\frac1n x^2$?

Chris Culter
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  • I did that with a line and it worked but then I thought there could be issues with mor complicated behaviour, like when we have piecewise defined functions, let me think maybe I was overthinking it. Like, I am sure sum of convex is convex, but is it strictly convex if one of the addends are strictly convex? – xyz Apr 15 '20 at 00:23
  • Yes it is I just proved it I think you are correct then thank you! – xyz Apr 15 '20 at 00:25