The name is absent

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 — c_j I Xj — c_j > 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.

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