  The present paper is the first in a series that addresses the calculation of
the full one-loop corrections of dark matter (DM) annihilation cross-sections
in the low mass region of the inert doublet model (IDM). We first review the
renormalisation of the model both in a fully on-shell (OS) scheme and a mixed
scheme combining on-shell (for the masses) and a $\overline{{\rm MS}}$ approach
when the partial invisible width is closed and does not allow the use of a full
OS scheme. The scale dependence introduced by the mixed scheme is shown to be
tracked through an analysis of a parametrisation of the tree-level
cross-section and the $\beta$ constant of a specific coupling. We discuss how
to minimise the scale dependence. The theoretical uncertainty brought by the
scale dependence leads us to introduce a new criterion on the perturbativity of
the IDM. This criterion further delimits the allowed parameter space which we
investigate carefully by including a host of constraints, both theoretical and
experimental, including in particular, new data from the LHC. We come up with a
set of benchmark points that cover three different mechanisms for a viable
relic density of DM: {\it i)} a dominance of co-annihilation into a fermion
pair, { \it ii)} annihilation into 2 vector bosons of which one is off-shell
that requires the calculation of a $2 \to 3$ process at one-loop, {\it iii)}
annihilation that proceeds through the very narrow standard model Higgs
resonance. Since the $2 \to 3$ vector boson channel features in all three
channels and is essentially a build up on the simpler annihilation to OS vector
bosons, we study the latter in detail in the present paper. We confirm again
that the corrected cross-sections involve a parameter that represents
rescattering in the dark sector that a tree-level computation in not sensitive
to.
