  Shielding mechanisms' influence on the ratio of perpendicular to parallel
scale lengths of multidimensional plasma electron hole equilibria are analyzed
theoretically and computationally. It is shown that the ``gyrokinetic'' model,
invoking perpendicular polarization, is based on a misunderstanding and cannot
explain the observational trend that greater transverse extent accompanies
lower magnetic field. Instead, the potential in the wings of the hole, outside
the region of trapped-electron depletion, has isotropic shielding giving
$\phi\propto {\rm e}^{-r/L}/r$, with the shielding length $L$ equal to the
Debye length for holes much slower than the electron thermal speed. Particle in
cell simulations confirm the analysis.
