  A central problem in discrete geometry, known as Hadwiger's covering problem,
asks what the smallest natural number $N\left(n\right)$ is such that every
convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of
at most $N\left(n\right)$ of its translates. Despite continuous efforts, the
best general upper bound known for this number remains as it was more than
sixty years ago, of the order of ${2n \choose n}n\ln n$.
  In this note, we improve this bound by a sub-exponential factor. That is, we
prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal
constant $c>0$.
  Our approach combines ideas from previous work by Artstein-Avidan and the
second named author with tools from Asymptotic Geometric Analysis. One of the
key steps is proving a new lower bound for the maximum volume of the
intersection of a convex body $K$ with a translate of $-K$; in fact, we get the
same lower bound for the volume of the intersection of $K$ and $-K$ when they
both have barycenter at the origin. To do so, we make use of measure
concentration, and in particular of thin-shell estimates for isotropic
log-concave measures.
  Using the same ideas, we establish an exponentially better bound for
$N\left(n\right)$ when restricting our attention to convex bodies that are
$\psi_{2}$. By a slightly different approach, an exponential improvement is
established also for classes of convex bodies with positive modulus of
convexity.
