# linear algebra – Existence of a hyperplane with strictly positive coefficients to contain an antichain in \$mathbb{Z}^n_+\$

Given a hyperplane $$alpha^T x = beta$$ in $$mathbb R^n$$, with $$beta > 0, alpha_i > 0$$ for all $$i in (n)$$. Then for any $${v^i} subseteq {x in mathbb Z^n_+ mid alpha^T x = beta}$$, it’s obvious to see that there must have: $${v_i}$$ forms an antichain with respect to the component-wise order. My question is, for a given set of less than $$n$$ positive integer vectors, to guarantee the existence of a hyperplane $$alpha^T x = beta$$ containing all of these integer points with $$alpha_ i > 0$$ for all $$i in (n)$$, is the antichain condition also sufficient?

Formally speaking:

Given an antichain $${v^i}_{i in(d)} subseteq mathbb Z^n_+$$ with $$d< n$$. (Here antichain is with respect to the component-wise order: for any $$i neq j in (d),$$ there exists $$t_1, t_2 in (n),$$ such that $$v^i_{t_1}>v^j_{t_1}, v^i_{t_2}.) Then: there exists a hyperplane $$alpha^T x = beta$$ containing all these integer points, and $$alpha_i > 0$$ for any $$i in (n)$$.

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