Packing dimension and Cartesian products

We show that for any analytic set A in R(d), its packing dimension dim, (A) can be represented as sup(B){dim(H)(A x B) - dim(H)(B)}, where the supremum is over all compact sets B in R(d), and dim, denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dim(P) (A) < d. In contrast, we show that the dual quantity inf(B){dim(P)(A x B) - dim(P) (B)}, is at least the ''lower packing dimension'' of A, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)