SAUSAGES ARE GOOD PACKINGS

Let B-d be the d-dimensional unit ball and, for an integer n, let C-n = {x(1), ..., x(n)} be a packing set for B-d, i.e., \x(i) - x(j)\ greater than or equal to 2, 1 less than or equal to i < j less than or equal to n. We show that for every rho < root 2 a dimension d(rho) exists such that, for d greater than or equal to d(rho), V(conv(C-n) + rho B-d) greater than or equal to V(conv(S-n) + rho B-d), where S-n is a ''sausage'' arrangement of n balls, holds. This gives considerable improvement to Fejes Toth's ''sausage'' conjecture in high dimensions. Further, we prove that, for every convex body K and rho < 1/32d(-2), V(conv(C-n) + rho K) greater than or equal to V(conv(S-n) + rho K), where C-n is a packing set with respect to K and S-n is a minimal ''sausage'' arrangement of K, holds.