  In this paper we prove the Kneser-Poulsen conjecture for the case of large
radii. Namely, if a finite number of points in Euclidean space $E^n$ is
rearranged so that the distance between each pair of points does not decrease,
then there exists a positive number $r_0$ that depends on the rearrangement of
the points, such that if we consider $n$-dimensional balls of radius $r>r_0$
with centers at these points, then the volume of the union (intersection) of
the balls before the rearrangement is not less (not greater) than the volume of
the union (intersection) after the rearrangement. Moreover, the inequality is
strict whenever the new point set is not congruent to the original one. Also
under the same conditions we prove a similar result about surface volumes
instead of volumes.
  In order to prove the above mentioned results we use ideas from tensegrity
theory to strengthen the theorem of Sudakov, R. Alexander and Capoyleas and
Pach, which says that the mean width of the convex hull of a finite number of
points does not decrease after an expansive rearrangement of those points. In
this paper we show that the mean width increases strictly, unless the expansive
rearrangement was a congruence.
  We also show that if the configuration of centers of the balls is fixed and
the volume of the intersection of the balls is considered as a function of the
radius $r$, then the second highest term in the asymptotic expansion of this
function is equal to $-M_n r^{n-1}$, where $M_n$ is the mean width of the
convex hall of the centers. This theorem was conjectured by Balazs Csikos in
2009.
