I have the following problem to consider using Fenchel's duality theorem. Compared to a few I've solved previously, which were very easy convex functions with linear equality constraints (so very straightforward conjugate functions and all the usual nice tricks such as completing the square etc. helped), this one seems like a bit of a curveball. The function is $$J(x)=\int_0^1 t^2x(t)dt\to\text{minimize}$$ subject to the set $$A=\left\{x\in L_2[0,1]:\int_0^1x(t)dt\ge\frac{1}{\sqrt{2}}||x||_{L_2},\int_0^1tx(t)dt=1\right\}$$ So, before I even get to the fact that the function is linear rather than convex in $x$, I first notice that since we are in $L_2$, doesn't $\int_0^1x(t)dt=||x||_{L_2}$? Thus, the first constraint seems rather trivial, only requiring that $x\neq 0$. Furthermore, just attempting to visualize somewhat the functions given (objective and constraint), does this function even admit a minimizer?
$\textbf{Edit}$- Sorry, just realized that actually, $x$ can in fact be zero, since it's not a strict inequality constraint, and thus it seems entirely trivial (of course just talking about the first constraint here and its logic not considering the second).