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}<v^j_{t_2}$.) Then: there exists a hyperplane $alpha^T x = beta$ containing all these integer points, and $alpha_i > 0$ for any $i in (n)$.