Cutting a d-simplex

Question

Why is it possible to get any possible subset of nodes of a d+1 simplex in IR^d using halfspaces?

Answer 1

For convenience, let the vertices of the simplex be $v_0 = (0,\ldots,0)$ and $v_i$ the vector with $(v_i)_i = 1$ and $(v_i)_j = 0$ otherwise. Let $y$ be a vector with entries in $\{0,1\}$. Then $v_i . y > 1/2$ for those $i \ge 1$ with $y_i = 1$, while $v_i . y < 1/2$ otherwise. Thus any set of vertices will be the set of vertices in either the halfspace $\{x : x . y < 1/2 \}$ or the halfspace $\{x: x . y > 1/2 \}$ (depending on whether the set contains $v_0$) for suitable $y$.