  The three-dimensional (3-D) shape of a galaxy inevitably is tied to how it
has formed and evolved and to its dark matter halo. Local extremely metal-poor
galaxies (XMPs; defined as having an average gas-phase metallicity < 0.1 solar)
are important objects for understanding galaxy evolution largely because they
appear to be caught in the act of accreting gas from the cosmic web, and their
3-D shape may reflect this. Here we report on the 3-D shape of XMPs as inferred
from their observed projected minor-to-major axial ratios using a hierarchical
Bayesian inference model, which determines the likely shape and orientation of
each galaxy while simultaneously inferring the average shape and dispersion. We
selected a sample of 149 XMPs and divided it into three sub-samples according
to physical size and found that (1) the stellar component of XMPs of all sizes
tends to be triaxial, with an intermediate axis approx 0.7 times the longest
axis and that (2) smaller XMPs tend to be relatively thicker, with the shortest
axis going from approx 0.15 times the longest axis for the large galaxies to
approx 0.4 for the small galaxies. We provide the inferred 3-D shape and
inclination of the individual XMPs in electronic format. We show that our
results for the intermediate axis are not clouded by a selection effect against
face-on XMPs. We discuss how an intermediate axis significantly smaller than
the longest axis may be produced by several mechanisms, including lopsided gas
accretion, non-axisymmetric star formation, or coupling with an elongated dark
matter halo. Large relative thickness may reflect slow rotation, stellar
feedback, or recent gas accretion.
