Packing dimension and Cartesian products

. We show that for any analytic set A in R d , its packing dimension dim P ( A ) can be represented as sup B f dim H ( A (cid:2) B ) − dim H ( B ) g ; where the supremum is over all compact sets B in R d , and dim H denotes Hausdor(cid:11) 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 f dim P ( A (cid:2) B ) − dim P ( B ) g ; 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 Hausdor(cid:11) dimension.)