Suppose I have a polyhedral complex $\{P_1, \ldots, P_k\}$ and let
$$S := \bigcup_{i = 1}^k P_i$$
I am interested in a function that 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? Thanks in advance!
Great question! I'm not sure if there is a definition of this specifically for polyhedra, but here is what I know from a convex analysis perspetive:
The "distance function to the set $C$" is denoted $d_C(x)=\inf_{c\in C}\|x-c\|$. There are some really neat results for when $C$ is nonempty, closed and convex. However, as soon as we restrict to considering the case when $C$ is the boundary of some other set, then $C$ is frequently nonconvex. e.g. the closed $\ell_1$ ball is is a convex polyhedron, but its boundary is nonconvex.