Let me state a standard result first. Let a $A\subset \mathbb{R}^d$ be a set of fixed volume. Define $A\_t$ to be the set of all points at distance at most $t$ from $A$. Then the volume of $A\_t$ is minimal if $A$ is a ball of the prescribed volume.


Another way to define $A\_t$ is by $A\_t=A+B(0,t)$, where $B(0,t)$ is the centered ball of radius $t$. We shall think of it as the union of translates of $A$ by all vectors in $B(0,t)$.


I am interested in extending such a result to the discrete setting. Say, we translate $A$ only in the $d$ orthogonal directions. That is, we look at the union $U(A)=\cup\_v (A+v)$, where $v$ is either the zero vector or $\pm e\_i$, where $e\_i$ is an element of the standard orthonormal basis.


Given that the volume of $A$ is fixed, which $A$ minimize the volume of $U(A)$? 



Unusual isoperimetry and maximizing the measure of unions of translates of a set
Fix large $N$.
Take $A$ to be union of $\varepsilon$-balls which centers have integer coordinates between $\pm N$. 
(You have to ajust $\varepsilon$ to get the needed volume.)


In this case $U(A)$ is a union $\varepsilon$-balls which centers have integer coordinates between $\pm (N+1)$.
Therefore $\mathrm{vol}[U(A)]$ can be made arbitrary close to $\mathrm{vol}A$ and there is no minimizer.


