Let $\mathcal{S}$ be an integer simplex of $\mathbb{R}^n$ with vertices $0, s_1, \ldots, s_n$ having exactly $k$ interior integer points and let $x = \sum_{i=1}^n t_i s_i$ be an interior integer point of $\mathcal{S}$. 18a. Show that $\sum_{i=1}^n t_i \leqslant 1 - \alpha(k,n)$. (One may reason by contradiction and construct then $k+1$ distinct integer points interior to $\mathcal{S}$.) 18b. Show that $\frac{\alpha(k,n)}{1-\alpha(k,n)} x \in (\mathcal{S} - x)$. 18c. Deduce that $a(\mathcal{S} - x) \geqslant \frac{\alpha(k,n)}{1-\alpha(k,n)}$.
Let $\mathcal{S}$ be an integer simplex of $\mathbb{R}^n$ with vertices $0, s_1, \ldots, s_n$ having exactly $k$ interior integer points and let $x = \sum_{i=1}^n t_i s_i$ be an interior integer point of $\mathcal{S}$.
18a. Show that $\sum_{i=1}^n t_i \leqslant 1 - \alpha(k,n)$. (One may reason by contradiction and construct then $k+1$ distinct integer points interior to $\mathcal{S}$.)
18b. Show that $\frac{\alpha(k,n)}{1-\alpha(k,n)} x \in (\mathcal{S} - x)$.
18c. Deduce that $a(\mathcal{S} - x) \geqslant \frac{\alpha(k,n)}{1-\alpha(k,n)}$.