Proof. Suppose that π ∈ Sn is such that the minimal j ∈ {1, . . . , n} for which aπ(j) ∈ BX(x, r+dX(x, y)) actually satisfies aπ(j) ∈ BX(x, r−dX(x, y)). Hence j r (x) = j and therefore aπ(j) = ar (x). Also, dX(aπ(j), y) 6 dX(aπ(j), x) + dX(x, y) 6 r, so j r (y) 6 j. But dX(x, ajπ r (y)) 6 dX(y, ajπ r (y)) + dX(x, y) 6 r+ dX(x, y), so by the definition of j we must have j r (y) > j. Thus j π r (...

The Political Parties, Elections and Referendums Act 2000 marked the most comprehensive and radical overhaul of British party finance for over 100 years. It instituted reforms in many areas with the notable exception of the comprehensive extension of state funding for political parties. However, despite the radicalism of the Act, questions have already arisen as to whether the further state fun...