  We apply the Alcock-Paczynski (AP) test to the stacked voids identified using
the large-scale structure galaxy catalog from the Baryon Oscillation
Spectroscopic Survey (BOSS). This galaxy catalog is part of the Sloan Digital
Sky Survey (SDSS) Data Release 12 and is the final catalog of SDSS-III. We also
use 1000 mock galaxy catalogs that match the geometry, density, and clustering
properties of the BOSS sample in order to characterize the statistical
uncertainties of our measurements and take into account systematic errors such
as redshift space distortions. For both BOSS data and mock catalogs, we use the
ZOBOV algorithm to identify voids, we stack together all voids with effective
radii of 30-100Mpc/h in the redshift range 0.43-0.7, and we accurately measure
the shape of the stacked voids. Our tests with the mock catalogs show that we
measure the stacked void ellipticity with a statistical precision of 2.6%. We
find that the stacked voids in redshift space are slightly squashed along the
line of sight, which is consistent with previous studies. We repeat this
measurement of stacked void shape in the BOSS data assuming several values of
Omega_m within the flat LCDM model, and we compare to the mock catalogs in
redshift space in order to perform the AP test. We obtain a constraint of
$\Omega_m = 0.38^{+0.18}_{-0.15}$ at the 68% confidence level from the AP test.
We discuss the various sources of statistical and systematic noise that affect
the constraining power of this method. In particular, we find that the measured
ellipticity of stacked voids scales more weakly with cosmology than the
standard AP prediction, leading to significantly weaker constraints. We discuss
how AP constraints will improve in future surveys with larger volumes and
densities.
