RangeVoting.org - Black's Singlepeakedness theorem & Condorcet Winner facts

Black's Singlepeakedness theorem & Condorcet Winner facts.
Executive Summary

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 L^p distance for each p≥1).
In 1D and all higher dimensions, with singlepeaked L^p-distance based utilities, CWs need not be the candidate with the highest social utility.
With utilities based on L² distance, for honest voters distributed centrosymmetrically, Condorcet voting methods are always optimum and in particular are superior to range voting. But with utilities based on L¹ distance – which is probably more realistic – Condorcet voting methods can deliver nonoptimal winners in ≥2 dimensions, and range voting appears to be superior to them.