In 1D a candidate located at the "median voter"
will be a CW (Condorcet "beats-all" winner).
But there need not be any candidate located at the median voter.
In 1D, with single-peaked utilities, a CW will always exist (at least up to ties).
But in 2D and all higher dimensions, CWs need not exist, and candidates at the
median voter need not
be CWs (for several meanings of "median" and with singlepeaked
utilities based on Lp distance for each p≥1).
In 1D and all higher dimensions, with singlepeaked Lp-distance based
utilities, CWs need not be the candidate with the highest social utility.
With utilities based on L² distance, for honest voters distributed
Condorcet voting methods are always optimum and in particular are superior
to range voting.
But with utilities based on
– which is probably more realistic –
Condorcet voting methods
can deliver nonoptimal winners in ≥2 dimensions,
and range voting appears to be superior to them.