# How to prove the following statement about convex hulls?

Consider $$M$$ finite sets of integer points $$P_m$$, $$m=1,\ldots,M$$. Let $$A = \left\{x_m\in\operatorname{conv}P_m, m=1,\dots,M, \sum_{m=1}^MN_mx_m=0\right\}$$ and $$B =\operatorname{conv}\left\{x_m\in P_m, m=1,\dots,M, \sum_{m=1}^MN_mx_m=0\right\}$$ where $$N_m$$ is a matrix of dimension compatible with $$x_m$$ and $$\operatorname{conv}P_m$$ is the convex hull of the points in $$P_m$$.

I was wondering if it is possible to prove that $$B\subseteq A$$ and how.

I think it should be possible.

Firstly, let us see if we can establish that $$A$$ is convex. Take

\begin{align}X &= (x_1,\ldots,x_M)\in A\\Y&=(y_1,\ldots,y_M)\in A.\end{align}

Let $$0\leq\lambda\leq 1$$. Then

$$\lambda X + (1-\lambda) Y = (\lambda x_1 + (1-\lambda)y_1,\ldots,\lambda x_M + (1-\lambda)y_M).$$

Since \begin{align}N_1x_1 + \ldots + N_Mx_M &= 0\\N_1y_1 + \ldots + N_My_M &= 0,\end{align} we have $$N_1(\lambda x_1 + (1-\lambda) y_1) +\ldots+ N_M(\lambda x_M + (1-\lambda) y_M) = 0.$$

Also, $$\forall i=1,\ldots,M, \lambda x_i + (1-\lambda) y_i\in\operatorname{conv}(P_i)$$.

Thus, convexity of $$A$$ is established.

$$B$$ is the convex hull of a subset of points of $$A$$. Let this subset be $$K\subseteq A$$.

So, $$B = \operatorname{conv}(K)\subseteq\operatorname{conv}(A)=A$$, since $$A$$ is convex.