convex polytopes – Distance to the “boundary” of a polyhedral complex


Suppose I have a polyhedral complex ${P_1, ldots, P_k}$ and let $S := cup_{i = 1}^k P_i$. I am interested in a function which measures the distance from a point $x in S$ to the “boundary” of my polyhedral complex, in other words the lower dimensional faces. More precisely, if $x in text{int}(P_i)$ for some $i$ then my function should return the distance from $x$ to the boundary of $P_i$. Otherwise, $x$ lies on a lower dimensional face and the function should return $0$.

Is there a name for this function? Has it appeared in the literature and been studied before? I’m also wondering if it has any special properties beyond simply being a distance function to a non-convex set. Thanks in advance!