  We consider spin-1/2 Fermi gases in arbitrary, integer or non-integer spatial
dimensions, interacting via a Dirac delta potential. We first generalize the
method of Tan's distributions and implement short-range boundary conditions to
arbitrary dimension and we obtain a set of universal relations for the Fermi
gas. Three-dimensional scattering under very general conditions of transversal
confinement is described by an effectively reduced-dimensional scattering
length, which we show depends on the three-dimensional scattering length in a
universal way. Our formula for non-integer dimensions interpolates between the
known results in integer dimensions 1, 2 and 3. Without any need to solve the
associated multichannel scattering problem, we find that confinement-induced
resonances occur in all dimensions different from D=2, while
reduced-dimensional contacts, related to the tails of the momentum
distributions, are connected to the three-dimensional contact by a correction
factor of purely geometric origin.
