  Let $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around
$x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$.
  In this survey paper we study results about the following type of problems:
How small can a set be if it contains a scaled copy of $T$ around every point
of a set of given size? We will consider the cases when $T$ is circle or sphere
centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square
centered at the origin, or more generally the $k$-skeleton ($0\le k<n$) of an
$n$-dimensional cube centered at the origin or the $k$-skeleton of a more
general polytope of ${\mathbb R}^n$.
  We also study the case when we allow not only scaled copies but also scaled
and rotated copies and also the case when we allow only rotated copies.
