We show that for any analytic set
A
A
in
R
d
\mathbf {R}^d
, its packing dimension
dim
P
(
A
)
\dim _{\mathrm {P}}(A)
can be represented as
sup
B
{
dim
H
(
A
×
B
)
−
dim
H
(
B
)
}
,
\; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} \, , \,
where the supremum is over all compact sets
B
B
in
R
d
\mathbf {R}^d
, and
dim
H
\dim _{\mathrm {H}}
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
\dim _{\mathrm {P}} (A) > d
. In contrast, we show that the dual quantity
inf
B
{
dim
P
(
A
×
B
)
−
dim
P
(
B
)
}
,
\; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} \, , \,
is at least the “lower packing dimension” of
A
A
, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)