Let $\phi$ be the family consisting of all random variables $X$ such that $P(X\in [0,1])=1$, $EX=\frac{1}{5}$, $P(X\leq\frac{3}{10})=\frac{1}{2}$. Calculate $\inf \{\rm{Var}(X):X \in \phi\}.$
Could you please give a hint? I have no idea how to start.
A sketch of a proof of Did's conjecture in a comment:
The contribution from $(\frac3{10},1]$ to the variance must be at least $\frac12(\frac3{10}-\frac15)^2=\frac12\cdot\frac1{100}$. If the probability in $[0,\frac3{10}]$ is concentrated in a Dirac mass, it also contributes at least $\frac12(\frac1{10}-\frac15)^2=\frac12\cdot\frac1{100}$, and otherwise even more. I don't know the name of the theorem used in that last statement, but it's analogous to the Huygens–Steiner theorem for moments of inertia.