  The claim that the W boson mass might be measured at the LHC with a precision
of O(10 MeV) is critically reviewed. It is argued that such a precision cannot
be achieved, unless a dedicated measurement program, specific to the LHC is
pursued. We propose such a program. Its main target is to significantly improve
the experimental control of the relative polarisation of the W+, W- and Z
bosons. We propose to achieve this goal either by running dedicated isoscalar
beams at the LHC or by running, in parallel to the standard p-p collision
program, a dedicated muon scattering "LHC-support-experiment" at the CERN SPS.
One of these auxiliary measurements is necessary for the "precision measurement
program" at the LHC, but not sufficient. It must be followed by dedicated
measurement strategies which are robust with respect to both the systematic
measurement uncertainties and to the perturbative and non-perturbative QCD
effects. We propose such strategies and evaluate their precision. At the LHC,
contrary to the Tevatron case, both the masses of the W+ and of the W- bosons
must be measured with high precision. In this context, we propose and evaluate
LHC dedicated strategies to measure the difference of the masses of the W+ and
W- bosons and of the absolute mass of the W boson assuming both masses are
equal. We show how one can overcome the obstacles in measuring the masses of W+
and W- to a precision of 10 MeV. We present a detailed evaluation of the
precision of the proposed methods based on the studies of a large, O(10^11),
sample of simulated W and Z production events.
