  For cross section measurements, an accurate knowledge of the integrated
luminosity is required. The FCC-ee Z lineshape programme sets the ambitious
precision goal of $10^{-4}$ on the \emph{absolute} luminosity measurement and
one order of magnitude better on the \emph{relative} measurement between
energy-scan points. The luminosity is determined from the rate of small-angle
Bhabha scattering, $\mathrm{e^+e^- \to e^+e^-}$, where the final state
electrons and positrons are detected in dedicated monitors covering small
angles from the outgoing beam directions. The constraints on the luminosity
monitors are multiple: \mbox{\emph{i}) they} are placed inside the main
detector volume only about 1\,m from the interaction point; \mbox{\emph{ii})}
they are centred around the outgoing beam lines and do not satisfy the normal
axial detector symmetry; \mbox{\emph{iii})} their coverage is limited by the
beam pipe, on the one hand, and the requirement to stay clear of the main
detector acceptance, on the other; \mbox{\emph{iv})} the steep angular
dependence of the Bhabha scattering process imposes a geometrical precision on
the acceptance limits at about 1\,\textmugreek rad, corresponding to
geometrical precisions of $\mathcal{O}(1\,\text{\textmugreek m})$; and
\mbox{\emph{v})} the very high bunch crossing rate of 50\,MHz during the Z-pole
operation calls for fast readout electronics. %Some constraints may be hard to
satisfy simultaneously. As an example, the high readout rate may lead to an
elevated level of heat dissipation rendering it difficult to maintain the
required geometrical stability. Inspired by second-generation LEP luminosity
monitors, a proposed ultra-compact solution is based on a sandwich of
tungsten-silicon layers. A vigorous R\&D programme is needed in order to ensure
that such a solution satisfies the more challenging FCC-ee requirements.
