  The length scale separation in dilute quantum gases in quasi-one-dimensional
or quasi-two-dimensional traps has spatially divided the system into two
distinct regimes. Whereas universal relations defined in strict one or two
dimensions apply in a scale that is much larger than the characteristic length
of the transverse confinements, physical observables in the short distances are
inevitably governed by three-dimensional contacts. Here, we show that p-wave
contacts defined in different length scales are intrinsically connected by a
universal relation, which depends on a simple geometric factor of the
transverse confinements. While this universal relation is derived for one of
the p-wave contacts, it establishes a concrete example of how dimensional
crossover interplays with contacts and universal relations for arbitrary
partial wave scatterings.
