Consider a triangulation $\mathcal{T}$ of a PL $n$-manifold. Suppose we have some property (e.g. a measure of complexity) which holds "locally" - for example, on neighbourhoods of the vertices of $\mathcal{T}$. Can anything be said/done to extend this local property to a "global" statement about the whole of $\mathcal{T}$? I imagine this is an open problem, so would be interested just to hear people's thoughts/ideas, possible ways to approach this, etc. Naively maybe one could hope to argue something like "Property X holds on a neighbourhood of the vertices (i.e. holds on $\mathrm{St}(v)$, for each vertex $v$ of $\mathcal{T}$), and so extending linearly over the simplices of $\mathcal{T}$, X holds on $\mathcal{T}$" - or something like this...