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!