Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex?
Edit: A helpful question that is maybe easier: Can every square-integrable function: $[0,1] \to \mathbb{R}$ be written as a linear combination of convex functions? If not, then of course there is no hope. If so, then I think the next thing I'd check would be if Gram-Schmidt can preserve the convexity of functions....
Edit #2: If it is easier, how about for the space of absolutely continuous functions from $[0,1]$ to $\mathbb{R}$? (Or is this space no longer complete, so the question no longer makes sense?)