Throughout this question, $\mathcal{S}$ is a simplex in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{S}}$. We want to show that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant 2\left\lfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{a(\mathcal{S})+1}\right)^n \right\rfloor + 1$$ We then set $a = a(\mathcal{S})$, and $k = \rfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a}{a+1}\right)^n \lfloor$.
10a. Express, for $\beta \in \mathbb{R}^*$ and $x \in \mathbb{R}^n$, $\operatorname{Vol}(\beta \mathcal{S})$ and $\operatorname{Vol}(\mathcal{S} - x)$. Show that for $\lambda \in [0,1[$ sufficiently close to $1$, $\operatorname{Vol}\left(\frac{\lambda a}{a+1}\mathcal{S}\right) > k$.
10b. For $\lambda$ as in the previous question, let $v_0, \ldots, v_k$ be the $k+1$ distinct points in $\frac{\lambda a}{a+1}\mathcal{S}$ satisfying $v_i - v_j \in \mathbb{Z}^n$ for all $i, j$, whose existence is guaranteed by Theorem 1. Show that the points $v_i - v_j$ are in $\lambda \mathcal{S}$. Deduce that the $v_i - v_j$ are in $\mathring{\mathcal{S}}$.
10c. Show that there exists an index $j \in \{0, \ldots, k\}$ such that the $(2k+1)$ points $0, \pm(v_i - v_j)$, for $i \in \{0, \ldots, k\} \setminus \{j\}$ are distinct. Deduce the statement of question 10, then that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{2}\right)^n$$