From Lorentz Invariance Violation to the preferred frame of reference I've read here and there that Lorentz Invariance Violation (LIV) would imply the existence of a preferred frame, a frame where some physical laws are valid, while invalid in other frames.
The fact there is no preferred frame is another way to express Einstein's principle of relativity (I stick to special relativity here), which is a key ingredient to derive Lorentz transformations, but I don't understand the logic allowing to go from LIV to the non-validity of the relativity principle. Could somebody help me understanding this, please ?
 A: The logic is straightforward. If the Lorenz invariance did not hold true, then the speed of light would not be constant in all inertial reference frames.
A: The essence of the Lorentz invariance lies in the invariance of the metric among (Cartesian) inertial frames which are related to one another via a constant velocity transformation (i.e., a boost-transformation). The violation of the Lorentz invariance means that there exists a frame $O$ which is related via a boost-transformation to another frame $P$ where the metric reads $ds^2=c^2 dt^2-dx^2$ but the metric in the frame $O$ cannot be chosen to read the same. This immediately implies that the speed of light (i.e., the maximum speed limit) is different than $c$ in $O$. But, the Maxwell laws imply that the speed of light must be $c$ and thus, they cannot be true in frame $O$. Thus, we immediately obtain a preferred frame of reference $P$ (or, a set of preferred frames of references) which we can experimentally distinguish via checking if the Maxwell laws hold in a given frame of reference or not.

Of course, the Maxwell laws hold no central place in the structure of the Lorentz invariance itself nor are they uniquely needed to prove that the violation of the Lorentz invariance means that the symmetry among inertial frames breaks down. I simply chose them to exemplify the statement which is actually self-evident at the point where we realize that the violation of the Lorentz invariance means that the metric doesn't remain invariant (among inertial frames) and that it directly implies different speeds of causality in different frames.
A: 
I've read here and there that Lorentz Invariance Violation (LIV) would imply the existence of a preferred frame, a frame in which physical laws would have a simpler form.

If your question comes as a consequence of the definition in Wikipedia, then I don't agree with this definition. A frame in which the laws of physics look simpler can exist in full agreement with the relativity.
See an example out of many:
Assume a source of quantum particles that emits in the direction perpendicular to a perfectly reflecting mirror. Interference fringes are produced near the mirror, maxima and minima. But, assume we look from a frame of reference in movement with respect to the source and mirror. The wave traveling to the mirror will appear as having a different frequency that the one returning. Though, the traveler will see near the mirror maxima and minima. And they are produced by waves of different frequencies. Of course the mathematical treatment of such a situation is more complicated, and we would like to return to treat it in the frame at rest with respect to the mirror.
So, no! The idea of preferred frames was introduced to mean that the some physical laws are valid there, and invalid in another frames.
An example: in base of the GHZ experiment (Greenberger-Horne-Zeilinger) a version of it can be built that can show that predictions about measurement results made in one frame of reference, contradict predictions made according to another frame of reference. That, because the wave-function in the two frames is different. If we want to make the two sets of predictions identical, we have to admit that in one of these frames the wave-function is not obeyed. (I gave you this example without getting into details of contextuality, and other more professional and complicated things).
