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][1].  Here the intersection of the boundary facets isn't even bounded.

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

  [1]: http://brownsharpie.courtneygibbons.org/images/counterexample.pdf