  We define the three-body scattering hypervolume $D_F$ for identical
spin-polarized fermions in two dimensions, by considering the wave function of
three such fermions colliding at zero energy and zero orbital angular momentum.
We derive the asymptotic expansions of such a wave function when three fermions
are far apart or one pair and the third fermion are far apart, and $D_F$
appears in the coefficients of such expansions. For weak interaction
potentials, we derive an approximate formula of $D_F$ by using the Born
expansion. We then study the shift of energy of three such fermions in a large
periodic area due to $D_F$. This shift is proportional to $D_F$ times the
square of the area of the triangle formed by the momenta of the fermions. We
also calculate the shifts of energy and of pressure of spin-polarized
two-dimensional Fermi gases due to a nonzero $D_F$ and the three-body
recombination rate of spin-polarized ultracold atomic Fermi gases in two
dimensions.
