TITLE: How to prove $ E(|X-Y|) \le E(|X+Y|)$ when $X,Y$ are i.i.d variables?
QUESTION [24 upvotes]: Let $X$ and $Y$ be independent random variables having the same distribution and the finite mathematical expectation. How to prove the inequality $$ E(|X-Y|) \le E(|X+Y|)?$$

REPLY [4 votes]: Let's consider the question of when $E[f(X,Y)] \geq 0$ in the generality of real-valued functions of arbitrary i.i.d. random variables on probability spaces.  With no loss of generality take $f$ to be symmetric in $X$ and $Y$, because $E[f]$ is the same as $E$ of the symmetrization of $f$.
There is a simple, and greatly clarifying, reduction to the case of random variables with at most two values.  The general case is a mixture of such distributions, by representing the selection of $(X,Y)$ as first choosing an unordered pair according to the induced distribution on those, and then the ordered pair conditional on the unordered one (the conditional distribution is the $1$ or $2$-valued distribution, and the weights in the mixture are the probability distribution on the de-ordered pair).  One then sees, after some more or less mechanical analysis of the 2-valued case, that the key property is:
$f(x,y)=|x+y| - |x-y|$, the symmetric function for which we want to prove $E[f(X,Y)] \geq 0$, is diagonally dominant. That is, $f(x,x)$ and $f(y,y)$ both are larger than or equal to $|f(x,y)|$.  By symmetry we really need only to check one of those conditions, $\forall x,y \hskip4pt f(x,x) \geq |f(x,y)|$.
A function satisfying these conditions, now on a general probability space, has non-negative expectation in the 2-valued case, because for $p+q=1$ (the probability distribution), $$E[f] = p^2 f(a,a) + q^2 f(b,b) + 2pq f(a,b) \geq (p-q)^2|f(a,b)| \geq 0$$ 
The equality cases when expectation is zero are when $p=q$ and $f(a,b) = -f(a,a) = -f(b,b)$.  For 1-valued random variables, equality holds at values where $f(p,p)=0$. Due to diagonal dominance these are null points, with $f(p,x)=0$ for all $x$.  
This allows a generalization and proof of Ewan Delanoy's observation, in the general situation: if the support of the random variable has an involution $\sigma$ such that $\sigma(p)=p$ for null points and for non-null points $b=\sigma(a)$ is the unique solution of $f(a,a)=f(b,b)=-f(a,b)$, then the expectation is zero (when finite) if and only if the distribution is $\sigma$-invariant.  That is because the expectation zero case must be a mixture of the $1$ and $2$-atom distributions with zero expectation, and all of those assign probability in a $\sigma$-invariant way to the atoms. 
Returning to the original problem, for $f(x,y)=|x+y| - |x-y|$ with the absolute value interpreted as any norm on any vector space, diagonal dominance follows from the triangle inequality, $0$ is a unique null point, and the involution pairing every non-null $x$ with the unique solution of $f(x,y)=-f(x,x)=-f(y,y)$ is $x \to -x$.  This recovers the characterization that the distribution is symmetric in the critical case, for any $f$ derived from a norm.
Note (*). In passing between ordered and unordered pairs, there might be some issue of "measurable choice" on general measure spaces, or not, and it is an interesting matter what exactly is true about that and whether any condition is needed on the measure space.  In the original problem one has a selection function  $(\min(X,Y),\max(X,Y))$, if needed to avoid any complications, and the same would be true in any concrete case by using order statistics on coordinates.