  The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if $A
\subset {\Bbb R}^d$ has positive Lebesgue measure than $A-A=\{x-y: x,y \in A\}$
contains an open ball. We obtain some quantitative lower bounds on the size of
this ball and in some cases, relate it to natural geometric properties of
$\partial A$. We also study the process $K_n =\frac{1}{2}(K_{n-1} - K_{n-1})$
when $K_0$ is a compact subset of $\mathbb{R}^d$ and determine various aspects
of its convergence to $Conv(K_1)$, the convex hull of $K_1$. We discuss some
connections with convex geometry, Weyl tube formula and the Kakeya needle
problem.
  \noindent
  {\it Keywords: Measure theory, Steinhaus theorem, Convex geometry, Weyl tube
formula.}
  \noindent
  2020 {\it Mathematics Subject Classification:} Primary: 28A75, 52A27.
Secondary: 52A30, 53A07.
