A quantitative version of the Steinhaus theorem

The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if $A\subset {\mathbb{R}}^d$ 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 $\mathrm{\partial }A$. 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.