  The densest local packing (DLP) problem in d-dimensional Euclidean space Rd
involves the placement of N nonoverlapping spheres of unit diameter near an
additional fixed unit-diameter sphere such that the greatest distance from the
center of the fixed sphere to the centers of any of the N surrounding spheres
is minimized. Solutions to the DLP problem are relevant to the realizability of
pair correlation functions for packings of nonoverlapping spheres and might
prove useful in improving upon the best known upper bounds on the maximum
packing fraction of sphere packings in dimensions greater than three. The
optimal spherical code problem in Rd involves the placement of the centers of N
nonoverlapping spheres of unit diameter onto the surface of a sphere of radius
R such that R is minimized. It is proved that in any dimension, all solutions
between unity and the golden ratio to the optimal spherical code problem for N
spheres are also solutions to the corresponding DLP problem. It follows that
for any packing of nonoverlapping spheres of unit diameter, a spherical region
of radius less than or equal to the golden ratio centered on an arbitrary
sphere center cannot enclose a number of sphere centers greater than one more
than the number that can be placed on the region's surface.
