Double vertical bar notation

Question

A common notation used for information divergence is D(P||Q). I hope to get something like D(S||Q) where S=||P||/2. I wonder if it is possible to display it nicely ? In such a way that the middle double bar has the right height. I tried

$D left( frac{|P|}{2} right| left| Q right)$

The above do not work as the middle bar are not of the same vertical height but its closest to what I hope to achieved. Using just | instead of right| left| for the middle double bar results in a short middle double bar.

Does anyone has a good idea on how to achieve that. Thanks
Else, I could just create a new notation.

-- Update -- 
I managed to find out how to do it with this useful operator middle

$D left( frac{|P|}{2} middle| Q right)$

Hope someone else may find this useful.

egreg · Accepted Answer

This double bar should be treated as a binary relation, so it will be distinguishable from other usages, such as a norm.

The simplest way to cope with the problem is to use mathtools facilities:

documentclass{article}
usepackage{mathtools}

DeclarePairedDelimiterX{infdivx}[2]{(}{)}{%
  #1;delimsize|;#2%
}
newcommand{infdiv}{Dinfdivx}
DeclarePairedDelimiter{norm}{lVert}{rVert}

begin{document}

begin{gather}
Dleft(P middle| Qright) rlap{quaditshape WRONG!} 
% normal size
infdiv{P}{Q} 
% fixed size delimiters
infdiv[bigg]{frac{norm{P}}{2}}{Q} 
% extensible delimiters
infdiv*{frac{norm{P}}{2}}{Q}
end{gather}

end{document}

Note that number 1 is wrong in two respects: there is an unwanted space between the "D" and the left parenthesis; the two bars are too near the symbols, making for ambiguity.

samerivertwice · Answer

In MathJax use $lVert$ and $rVert$.  The capital v gives you double bars while the lower case v in $lvert$ gives you single vertical bars.  But it's not clear to me what to you is the "right" height.
Also, you have $vert$ and $Vert$.
Since you mention using a vertical bar as a binary relation, you may want $mid$ which is kerned to sit midway between its neighbours.  But that has no double-bar version so finally, you have the option of $mathrel{Vert}$ which is both double-bar and kerned to the centre.

Double vertical bar notation

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