Lemma: Uniform convex filling

If $L={\max }_{s\in S}\Vert s{\Vert }_2$, then

for every integer $k\ge 0$.

Proof. Symmetry gives $0\in C_S$, proving the first inclusion. For $z\in k{C}_S$, Caratheodory's theorem gives $z/k={{\sum }_{i=1}}^{D+1}{\lambda }_i{s}_i$, with nonnegative weights summing to one. This representation follows by eliminating positive weights along affine dependences until at most $D+1$ remain. Set $m_i=\lfloor k{\lambda }_i\rfloor$ and $y={\sum }_i{m}_i{s}_i$. Then ${\sum }_i{m}_i\le k$, so $y\in B_S(k)$, and

The case $k=0$ is immediate.