  We consider the traveling salesman problem when the cities are points in R^d
for some fixed d and distances are computed according to geometric distances,
determined by some norm. We show that for any polyhedral norm, the problem of
finding a tour of maximum length can be solved in polynomial time. If
arithmetic operations are assumed to take unit time, our algorithms run in time
O(n^{f-2} log n), where f is the number of facets of the polyhedron determining
the polyhedral norm. Thus for example we have O(n^2 log n) algorithms for the
cases of points in the plane under the Rectilinear and Sup norms. This is in
contrast to the fact that finding a minimum length tour in each case is
NP-hard. Our approach can be extended to the more general case of quasi-norms
with not necessarily symmetric unit ball, where we get a complexity of
O(n^{2f-2} log n).
  For the special case of two-dimensional metrics with f=4 (which includes the
Rectilinear and Sup norms), we present a simple algorithm with O(n) running
time. The algorithm does not use any indirect addressing, so its running time
remains valid even in comparison based models in which sorting requires Omega(n
\log n) time. The basic mechanism of the algorithm provides some intuition on
why polyhedral norms allow fast algorithms.
  Complementing the results on simplicity for polyhedral norms, we prove that
for the case of Euclidean distances in R^d for d>2, the Maximum TSP is NP-hard.
This sheds new light on the well-studied difficulties of Euclidean distances.
