  The sphere packing problem asks for the densest packing of unit balls in
d-dimensional Euclidean space. This problem has its roots in geometry, number
theory and it is part of Hilbert's 18th problem. In 1958 C. A. Rogers proved a
non-trivial upper bound for the density of unit ball packings in d-dimensional
Euclidean space for all d>0. In 1978 Kabatjanskii and Levenstein improved this
bound for large d. In fact, Rogers' bound is the presently known best bound for
43>d>3, and above that the Kabatjanskii-Levenstein bound takes over. In this
paper we improve Rogers' upper bound for the density of unit ball packings in
Euclidean d-space for all d>7. We do this by estimating from below the surface
area of Voronoi cells in any packing of unit balls in Euclidean d-space for all
d>7.
