  In this paper we prove a theorem that provides an upper bound for the density
of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this
theorem is a generalization of the linear programming bound for sphere
packings. We illustrate its use by computing an upper bound for the maximum
density of packings of regular pentagons in the plane. Our computational
approach is numerical and uses a combination of semidefinite programming, sums
of squares, and the harmonic analysis of the Euclidean motion group. We show
how, with some extra work, the bounds so obtained can be made rigorous.
