Suppose that $T$ and $S$ are two $(d + 1)$-element sets in $\mathbb{R}^d$ such that $o\in conv(x : x\in T)$ and $o\in conv(x : x\in S)$. Prove that there is a sequence of $(d + 1)$-element sets $S_0 = T, S1, . . . , S_{d+1} = S$ such that $S_i\subset S\cup T$, $S_i\cap S_{i−1}$ is a $d$-element set, $o\in conv(x\in S_i)$ for any $i = 1, . . . , d + 1$.
Comment: It is an exercise in chapter 8 of the book "Lectures on discrete geometry" by Matousek. It seems we should start by $T$ and show that we can replace one of its points by a point of $S$ in a way that $0$ still remain in the convex hull of a new set and so on. I just proved we can find a subset of $S^{'}\subset {S}$ and $T^{'}\subset T$ such that $|S^{'}|+ |T^{'}|\leq d+2$ and $conv (S^{'})\cap conv(T^{'})\neq\emptyset$. But I am not sure to can finish the proof in this way.
