Absolute value inequality for complex numbers I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality 
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
hold for all complex numbers $a,b,c$ ?
For real values a case analysis will verify the inequality. 
What is desired is a proof using the triangle inequality or a counterexample.
Thanks in advance.   
 A: In general, once you've proven an inequality like this in ${\bf R}$
it holds automatically in any Euclidean space (including ${\bf C}$)
by averaging over projections.  ("Inequality like this" = inequality
where every term is the length of some linear combination of
variable vectors in the space; here the vectors are $a,b,c$.)
In the case of complex numbers we have
$$
|z| = \frac14 \int_0^{2\pi} \bigl| {\rm Re}(e^{i\theta} z) \bigr| \, d\theta.
$$
Applying this to $z=a$, $b$, $c$, and $a \pm b \pm c$ reduces
the desired inequality to the one-dimensional case.  In $d$-dimensional
space we'd write $C\|z\|$ as an average of $|u \cdot z|$ over $u$ in the
unit sphere (for a suitable constant $C>0$).
I learned this trick at
MOP
30+ years ago, and don't know or remember who discovered it.
I didn't even know that the specific inequality we were assigned
was due to Hlawka (if I remember right that it was the inequality
$$
 \|x+y\| + \|y+z\|+\|z+x\| \le \|x\|+\|y\| + \|z\| + \|x+y+z\|
$$
quoted by Suvrit).  We were shown the averaging solution
after laboring to prove it bare-handed.  The reference Suvrit cites
does not use the averaging method, so I do not know whether it too
is due to Hlawka or to another mathematician.
A: It seems that your inequality is just an incarnation of Hlawka's inequality 
which says that for any vectors $x, y, z$ in an inner product space $V$ we have
\begin{equation*}
 \|x+y\| + \|y+z\|+\|z+x\| \le \|x\|+\|y\| + \|z\| + \|x+y+z\|.
\end{equation*}
Using $x=a+b-c$, $y=a+c-b$, and $z=b+c-a$ we obtain the inequality in the OP.

Additional remarks:
To add some more context, please see the paper linked here, which provides quite a nice summary of work related to Hlawka's inequality, which apparently stems back to a 1942 paper of Hornich (also cited by Zurab below). The paper linked to above explores the interesting generalization:
\begin{equation*}
 f(x+y) + f(y+z) + f(z+x) \le f(x+y+z) + f(x)+f(y)+f(z),
\end{equation*}
where $x,y,z$ may come from an Abelian group, or a linear space, or the real line---each with its own set of conditions on the mapping $f$. The functional form of Hlawka's inequality is credited to a 1978 paper of Witsenhausen.
A: In fact the Hlawka's inequality first appeared (as a special case of more general result) in H. Hornich, Eine Ungleichung für Vektorlängen, Mathematische Zeitschrift 48 (1942), 268-274 http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0048&DMDID=DMDLOG_0025&LOGID=LOG_0025&PHYSID=PHYS_0256
(see p. 268. P.S. as Joni Teräväinen has remarked, Hornich credits on page 274 to Hlawka an algebraic proof of this special case and reproduces it).
Hlawka's original proof, besides the book indicated by Suvrit, can be found in "Classical and New Inequalities in Analysis" by D.S. Mitrinovic, J. Pecaric and A.M Fink, p. 521 and in "Analytic Inequalities" by D.S. Mitrinovic, p.171. Both books provide Adamovic and Djorkovic generalizations of the Hlawka's inequality.
Interestingly, all these generalizations are special cases of more general result given in http://www.sciencedirect.com/science/article/pii/S0022247X96904588 (Generalizations of Dobrushin's Inequalities and Applications, by M. Radulescu and S. Radulescu).
Another proof of Hlawka's inequality can be found in http://www.sbc.org.pl/Content/34160/1995_13.pdf (On two geometric inequalities, by A. Simon, P.
Volkmann), and still another one in http://www.jstor.org/discover/10.2307/2310890?uid=3738936&uid=2&uid=4&sid=21104051771107 (The Polygonal Inequalities, by D.M. Smiley and M.F. Smiley).
