Suppose you have an equidimensional $n$-dimensional simplicial complex $\Delta \subseteq \mathbb Q^n$; i.e., $\Delta$ is the union of finitely many $n$-simplices that intersect only along proper faces. (I really do mean to use the same $n$.) By an $n$-simplex, I mean the convex hull of $n$ affinely independent points in $\mathbb Q^n$. In this setup, a boundary facet is any $(n-1)$-simplex that is a facet of exactly one of the $n$-simplices that make up $\Delta$. Each boundary facet lies on a unique hyperplane, and the $n$-simplex to which it belongs lies entirely on one halfspace.

I'm having trouble proving the geometrically reasonable (maybe even obvious!) claim that the intersection of these halfspaces is contained in $\Delta$.

Question: Is it true that the intersection of these halfspaces is contained in $\Delta$? If so, can you point me to a reference? I couldn't find a proof of this in Ziegler's Lectures on Polytopes.

One thing that is throwing me off is that the claim isn't true if $\Delta$ isn't of full dimension. For instance, consider the situation in this picture: counterexample picture. Here the intersection of the boundary facets isn't even bounded.

I'm not a discrete mathematician, so thanks for bearing with me.

Suppose you have an equidimensional $n$-dimensional simplicial complex $\Delta \subseteq \mathbb Q^n$; i.e., $\Delta$ is the union of finitely many $n$-simplices that intersect only along proper faces. (I really do mean to use the same $n$.) By an $n$-simplex, I mean the convex hull of $n+1$ affinely independent points in $\mathbb Q^n$. In this setup, a boundary facet is any $(n-1)$-simplex that is a facet of exactly one of the $n$-simplices that make up $\Delta$. Each boundary facet lies on a unique hyperplane, and the $n$-simplex to which it belongs lies entirely on one halfspace.

I'm having trouble proving the geometrically reasonable (maybe even obvious!) claim that the intersection of these halfspaces is contained in $\Delta$.

Question: Is it true that the intersection of these halfspaces is contained in $\Delta$? If so, can you point me to a reference? I couldn't find a proof of this in Ziegler's Lectures on Polytopes.

One thing that is throwing me off is that the claim isn't true if $\Delta$ isn't of full dimension. For instance, consider the situation in this picture: counterexample picture. Here the intersection of the boundary facets isn't even bounded.

I'm not a discrete mathematician, so thanks for bearing with me.