The name is absent

10

The geometric context of the next theorem can be easily understood in two
dimension.

Theorem 3.5 Exclusion Theorem

Let a simplex P be given in standard form with the additional condition that
ff₁≠i ∣^t¾∣ < ^aa holds for all i = 1,    , n. Then the Labeling Rule precludes the

possibility of the coexistence of a completely labeled simplex of type I and a com-
pletely labeled simplex of type II. If P contains an integral point, then there exists
no completely labeled simplex of type II.

Proof: We only need to consider the case in which P contains an integral point,
say æ⁰, i.e., Ax⁰ ≤ b. Let us suppose to the contrary that there is a completely
labeled simplex of type II, say σ(⅛¹,π) with vertices x¹, ■■■, xⁿ⁺¹, where π =
(τr(l), ∙ ∙ ∙ , π(n ⅛ 1)) is a permutation of the n ⅛ 1 elments of N, and

x^t+1 = x^t ⅛ q(π(if), i = 1, ∙ ∙ ∙ , n;

x¹ = xⁿ⁺¹ + q(π(n + 1)).

Now it is easy to see that there exist nonnegative integers k^, ∙ ∙ ∙, such that

x¹ = x° + ∑ k⅛(i),
ieN

and

min kl = 0.
heN ⁿ

Let

I = argmin{ π ¹(∕z) ∣ kj_l = max: k¹_j

Then there exist nonnegative integers k^t₁, ■ ■ - , k^t_n+1 such that

x^t = x⁰ + ∑ k^t_jq(f)
j∈N

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