Let $\mathcal{S}$ be a simplex of $\mathbb{R}^n$ and $k$ an integer such that $\operatorname{Vol}(\mathcal{S}) > k$. 15a. Show that there exist $x \in [0,1[^n$ and $(k+1)$ elements of $\mathbb{Z}^n$ $u_0, \ldots, u_k$ such that $x \in \mathcal{S} - u_i$ for $i = 0, \ldots, k$. One may study the sets $(u + [0,1[^n) \cap \mathcal{S}$ when $u$ ranges over $\mathbb{Z}^n$; and admit — outside the CPGE curriculum — that the volume of a simplex is its Lebesgue measure, which is sub-additive. 15b. Deduce from this the existence of the $(k+1)$ points $v_0, \ldots, v_k$ that satisfy the conditions of Theorem 1. 15c. Prove Theorem 1, that is, here we assume only that $\operatorname{Vol}(\mathcal{S}) \geqslant k$.
Let $\mathcal{S}$ be a simplex of $\mathbb{R}^n$ and $k$ an integer such that $\operatorname{Vol}(\mathcal{S}) > k$.
15a. Show that there exist $x \in [0,1[^n$ and $(k+1)$ elements of $\mathbb{Z}^n$ $u_0, \ldots, u_k$ such that $x \in \mathcal{S} - u_i$ for $i = 0, \ldots, k$.
One may study the sets $(u + [0,1[^n) \cap \mathcal{S}$ when $u$ ranges over $\mathbb{Z}^n$; and admit — outside the CPGE curriculum — that the volume of a simplex is its Lebesgue measure, which is sub-additive.
15b. Deduce from this the existence of the $(k+1)$ points $v_0, \ldots, v_k$ that satisfy the conditions of Theorem 1.
15c. Prove Theorem 1, that is, here we assume only that $\operatorname{Vol}(\mathcal{S}) \geqslant k$.