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I am working on a problem where I need the following property that I guess should be true but I am not able to prove it.

I have a bounded convex function $F(x, y)$ on $X\times Y$ (Think of $X=Y=\mathbb{R}^n$). We know that $C_b(X)\times C_b(Y)$ is dense in $C_b(X\times Y),$ and convex functions are continuous, it follows that we can approximate $F(x, y)$ (in sup norm) by linear combination of functions of the form $f(x)g(y)\in C_b(X)\times C_b(Y).$

My question is the following: Can we assume $f(x)$ and $g(y)$ to be convex as well?

Improve this question
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    $\begingroup$ The linear span of the elementary tensors $f\otimes g: X\times Y\to\mathbb R$, $(x,y)\mapsto f(x) g(y)$ is dense, not the elementary tensors themselves. $\endgroup$
    – Jochen Wengenroth
    Feb 9, 2021 at 6:00
  • $\begingroup$ You are right, I will correct the question in the morning. What I meant in the question is that can we approximate $F$ by linear combination of $f(x)g(y)$ where $f, g$ are convex? $\endgroup$
    – Raghav
    Feb 9, 2021 at 9:36
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    $\begingroup$ I think you should remove the hint "Think of $X = Y = \mathbb{R}^n$". Each convex and bounded function on these spaces is constant. $\endgroup$
    – Dieter Kadelka
    Feb 9, 2021 at 9:49
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    $\begingroup$ If $X$ and $Y$ are compact subsets of $\mathbb{R}^n$ wlog in $[0,\infty)^n$, all monomials $x_i^p$ are convex and span a dense algebra per Weierstrass thm. $\endgroup$
    – Pietro Majer
    Feb 9, 2021 at 9:53
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    $\begingroup$ Did you notice that a tensor product $f(x)g(y)$ of convex function need not be convex ? Your question does not make sense. $\endgroup$
    – Denis Serre
    Feb 9, 2021 at 9:59

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