Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1].
Is there a known tight upper bound in the number of polytopes in $\mathcal{C}$ for $\sum_{P \in \mathcal{C}} f(P)$ where $f(P)$ denotes the number of faces of the polytope $P$?
Note: A trivial upper bound can be achieved by the dual version of the upper bound theorem [2].