SELF-PACKING OF CENTRALLY SYMMETRICAL CONVEX-BODIES IN R2

Let B be a compact convex body symmetric around 0 in R2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radius rho(m, B) is the smallest t such that tB can be packed with m translates of the interior of B. For m less-than-or-equal-to 6 we show that the self-packing radius rho(m, B) = 1 + 2/alpha(m, B) where alpha(m, B) is the Minkowski length of the side of the largest equilateral m-gon inscribed in B (measured in the Minkowski metric determined by B). We show rho(6, B) = rho(7, B) = 3 for all B, and determine most of the largest and smallest values of p(m, B) for m less-than-or-equal-to 7. For all m we have (m/delta(B))1/2 - 3/2 less-than-or-equal-to rho(m, B) less-than-or-equal-to (m/delta(B))1/2 + 1, where delta(B) is the packing density of B in R2.