That is, if $f$ is convex, is its approximation convex?

If yes, it can be used to solve in positive a question from the monograph of Deville, Godefroy and Zizler "Smoothness and Renormings in Banach Spaces", if an arbitrary norm on Hilbert space can be approximated by C^2 norm with arbitrary precision. (I can specify the page, but not right now, since I do not have the book near by.)

Article Real analytic approximations which almost preserve Lipschitz...

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