appendix: Cuts
pdf-file

STATUS OF CONJECTURES 4.1 (p. 38):

As said above, Conjecture 4.1.(c) is destroyed by the examples of Minfeng Wang.
Conjecture 4.1.(a) is also false. It is equivalent to the following statement:
* Any stratum, up to permutation, contains a unique
length vector **b*_{min}=(b_{1}, ... , b_{n})
(possibly a conventional representative) with integral coordinates and minimal perimeter.
A counterexample with *n=12* is given by the length vectors
*(1,3,5,6,8,11,12,23,28,31,31,38)* and *(1,3,5,7,8,11,12,23,28,31,31,37)*, which belong to the same chamber
and have both minimal perimeter (137). This counterexample was actually found in 1959 by
John R. Isbell
and appeared in his paper ''On the enumeration of majority games'', Math. Comp. 13 (1959) 21-28
(pdf-file).

Conjecture 4.1.(b) has been solved positively by
Matthias Franz. More precisely:

**Proposition.** * A length vector
**b*_{min} as in Conjecture 4.1.(a) is generic if and only if its
perimeter is an odd integer.

*Proof. *
Obviously, an integer vector with odd perimeter is generic.
It remains to show that a generic vector *v* with integral
coordinates is not of minimal perimeter if the latter is even.
Observe that the perimeter of *v * is the dot product of *v*
with the vector *(1,..., 1)*. This is congruent
*Mod 2* to the dot product *v⋅h* with any
vector *h = (+/-1, ..., +/-1)* orthogonal to an hyperplane delimiting the chamber.
Since *v* is generic with even coordinate sum, *v⋅h* must be non-zero and even.
Decreasing any non-zero coordinate of *v* by 1 changes each dot product
*v⋅h* only by *+/-1*. So this gives another vector with integral coordinates in the same
chamber, but with smaller perimeter.

PROGRAMS: