Assume that $A_1,A_2,\dots,A_{k}\subset\mathbb{R}^n$ are $k$ convex sets and $k\geq n+1$. Is it true that there exist two distinct points $\gamma_1,\gamma_2\in\mathbb{R}_{\geq 0}^k$ such that $$ \gamma_{11}A_1+\gamma_{12}A_2+\dots+\gamma_{1{k}}A_k=\gamma_{21}A_1+\gamma_{22}A_2+\dots+\gamma_{2{k}}A_k \quad? $$
Note that this claim holds when $A_i$ are point-sets. In this case, I may define a $n\times k$-matrix $$ A = \begin{bmatrix} A_1 & A_2 & \dots & A_k \end{bmatrix} $$ and since $k>n$ there exist $\gamma_1,\gamma_2$ such that $A\gamma_1=A\gamma_2$. It is also not hard to show that these $\gamma_i$ can be chosen from $\mathbb{R}_{\geq 0}^k$.
I thought some version of Helly's theorem should apply here, but I cannot see how.