maybe this is a stupid question, but I dare it anyway: Let $\Omega$ be some bounded domain in ${\mathbb R}^n$. Then under all $L^1(\Omega)$ functions $f$ of fixed $L^1$-norm one, the constant function $\frac 1 {|\Omega|} 1\_{\Omega}$ minimises the $L^2(\Omega)$ norm, as a quick Cauchy-Schwarz argument shows. 


Question: Is this the only minimizing function or are there others? 



minimal $L^2$ norm with $L^1$ norm fixed to one
There are lots of others -- think of the function equal to $1/|\Omega|$ on one half of $\Omega$, and to $-1/|\Omega|$ on the other half. However if you restrict to non-negative functions then it is the only minimizer, as the equality case in Cauchy-Schwarz shows.


