An n-simplex σ is called a completely labeled simplex if ∣Z(σ)∣ = n⅛ 1. Specifically,
an n-simplex σ is called a completely labeled simplex of type I if L(σ) = { 0 } U N_i
for an index i ∈ N. Whereas an n-simplex σ is called a completely labeled simplex
of type II if L(σ) = N. Observe that a completely labeled simplex of type I has a
vertex being an integral point in P.
Now we state our basic results.
Theorem 2.1 The Labeling Rule results in at least one completely labeled sim-
plex.
Proof: It can be derived by induction. We omit the details. □
Furthermore, one can derive the following sharper and more important results.
Theorem 2.2 If P does not contain any integral point, then the Labeling Rule
results in a unique completely labeled simplex.
Clearly, the unique completely labeled simplex must be of type II. A proof of the
above theorem is deferred to Section 4. This theorem can be seen as a generalization
of the following lemma (see van der Laan [4] and Talman [14] ).
Lemma 2.3 Choose an arbitrary point c ∈ Rn. We assign x ∈ Zn with the label
/(æ) = i if i is the smallest index for which
xi ~ ci = max{ Xj — cj I Xj — cj > 0, j ∈ ʌ }.
If Xi ≤ Ci for all i = 1, ■ ■ -, n, we assign x with the label /(æ) = n ⅛ 1. Then there
exists a unique completely labeled simplex.
Theorem 2.4 If P has an integral point, then the Labeling Rule results in at
least two completely labeled simplices. Moreover, there exists at most one completely
labeled simplex of type II.