  We analyze the collision of three identical spin-polarized fermions at zero
collision energy, assuming arbitrary finite-range potentials, and define the
corresponding three-body scattering hypervolume $D_F$. The scattering
hypervolume $D$ was first defined for identical bosons in 2008 by one of us. It
is the three-body analog of the two-body scattering length. We solve the
three-body Schr\"{o}dinger equation asymptotically when the three fermions are
far apart or one pair and the third fermion are far apart, deriving two
asymptotic expansions of the wave function. Unlike the case of bosons for which
$D$ has the dimension of length to the fourth power, here the $D_F$ we define
has the dimension of length to the eighth power. We then analyze the
interaction energy of three such fermions with momenta $\hbar\mathbf{k}_1$,
$\hbar\mathbf{k}_2$ and $\hbar\mathbf{k}_3$ in a large periodic cubic box. The
energy shift due to $D_F$ is proportional to $D_F/\Omega^2$, where $\Omega$ is
the volume of the box. We also calculate the shifts of energy and pressure of
spin-polarized Fermi gases due to a nonzero $D_F$ and the three-body
recombination rate of spin-polarized ultracold atomic Fermi gases at finite
temperatures.
