I have a basic combinatorial question about the $f$-vector $(f_{-1}, f_0, f_1, ..., f_{n-1})$ of a simplicial complex $\Delta\subseteq 2^{[n]}$, where $f_i = \#\{\text{faces in $\Delta$ of size $i+1$}\}$. Must it always be the case that $f_i \ge f_{n-i-1}$ for $i \le n/2$? In other words, is $\Delta$ always "bottom heavy" in the sense that the faces of dimension $i$ number at least as many as those of codimension $i$ for $i \le n/2$?
I'm asking because I would like to say that the "average size" of a face in $\Delta$ is no more than $n/2$—formally I would like $\sum_{\sigma\in \Delta} |\sigma| \le n|\Delta|/2$.