  We conduct a condition number analysis of a Hybrid High-Order (HHO) scheme
for the Poisson problem. We find the condition number of the statically
condensed system to be independent of the number of faces in each element, or
the relative size between an element and its faces. The dependence of the
condition number on the polynomial degree is tracked. Next, we consider HHO
schemes on cut background meshes, which are commonly used in unfitted
discretisations. It is well known that the linear systems obtained on these
meshes can be arbitrarily ill-conditioned due to the presence of sliver-cut and
small-cut elements. We show that the condition number arising from HHO schemes
on such meshes is not as negatively effected as those arising from conforming
methods. We describe how the condition number can be improved by aggregating
ill-conditioned elements with their neighbours.
