$f_0$ and $f_1$ are two continuous density functions on $\mathbb{R}$. I wonder if $$(x,y):=\Bigg(\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u}\Bigg)$$ for all $f_0$ and $f_1$ is complete for some $0<u<1$? I cannot imagine such a space for any $0<u<1$. For $u=0.5$, it is however not difficult to see that it is a line $y=x$, c.f., the comments. But I imagine that in two dimensional space $(x,y)$ there should be at least a closed space which is a subset of $[0,1]^2$ but I am not sure if there is a hole somewhere.
Thank you very much.