If $s_0 \lt s_1 \lt \dots \lt s_q$ are simplexes in some euclidean space, then $\{b^{s_0}, \dots, b^{s_q}\}$ is affine independent, where $b^{s_i}$ means the barycenter of the $s_i$ simplex and $s_1 \lt s_2$ means $\text{Vert($s_1$)} \subset \text{Vert($s_2$)}$.
I can see that I would have to show that: $$\sum _{i=1}^q c_i(b^{s_i}-b^{s_0})= 0 \Rightarrow c_i=0$$
But for all $i \lt q$, $$\text{Vert}(s_i) \subset \text{Vert}(s_q)$$ therefore every barycenter $b^{s_i}$ can be written as a linear combination of vertices in $s_q$.
Therefore:
$$\sum _{i=1}^q c_i(b^{s_i}-b^{s_0})= c_1(\sum(m_i^{s_1}-m_i^{s_0})v_i) + \dots + c_q(\sum(m_i^{s_q}-m_i^{s_0})v_i)$$ $$= c_1(\sum d_i^{s_1}v_i) + \dots + c_q(\sum d_i^{s_q}v_i), \text{ where $d_i^{s_j} = m_i^{s_j} - m_i^{s_0}$ } $$
Which can be rewritten as
$$(\sum c_id_1^{s_i}) v_1 + \dots +(\sum c_id_q^{s_i}) v_q= 0 \Rightarrow \text{ each coefficient $=0$ }$$
but from here I can't see any way to show that each $c_i$ would be $0$.
Anyone have any ideas?