Sorry, it's a difficult question to give a title to, so if someone can think of a better one then feel free. I am trying to figure out how to set up a problem.
Suppose we have two bounded, closed subsets $X, Y$ of a fixed $\mathbb{R}^n$ and we fix a point $x_0 \in X$, and consider a closed arc $A$ in $\mathbb{R}^n \setminus Y$ with left end point $x_0$ and right end point $x_1$ (the latter not necessarily in $X$). We may assume that $A$ is a smooth, or even analytic, embedding if it's helpful.
For each $a \in A$ let $f_a$ be the affine shift of $\mathbb{R}^n$ which sends $x_0$ to $a$. Thus we obtain a continuous family of maps on $X$ indexed by $[0,1]$ which will sweep out an area, when we restrict these to $X$.
This is what I want to calculate the measure of: $$\lbrace x \in X | f_a(x) \notin Y \text{ for any } a \in A \rbrace$$
In other words, I want to calculate the measure of the part of $X$ which never hits $Y$ as the space $X$ is dragged along the curve $A$. It is fine to assume that both are connected. How the heck do I set this up? Is there an averaging method (a la Bogolyubov-Krillov) we can use? It is also fine to assume that $X$ and $Y$ are embedded $n$-manifolds with boundary, so in particular you could alternatively assume that $X, Y$ are open, or regular closed, etc. In other words, they're nice, set it up any way you would like. Locally connected, simply connected, pretty much anything is fine.
What I would really like is an integral equation with inputs $X, Y, A$ that works at least for SOME scenario. Heck, you can assume that $X, Y$ are just cubes oriented along with the axes, then we could subdivide. Even the case $n = 2$ would be very useful.
Thanks a lot!