The Next Generation Virgo Cluster Survey. VII. The intrinsic shapes of low-luminosity galaxies in the core of the Virgo cluster, and a comparison with the Local Group
(Abridged) We investigate the intrinsic shapes of low-luminosity galaxies in the central 300 kpc of the Virgo cluster using deep imaging obtained as part of the NGVS. We build a sample of nearly 300 red-sequence cluster members in the yet unexplored $-14 < M_{g} < -8$ magnitude range. The observed distribution of apparent axis ratios is then fit by families of triaxial models with normally-distributed intrinsic ellipticities and triaxialities. We develop a Bayesian framework to explore the posterior distribution of the model parameters, which allows us to work directly on discrete data, and to account for individual, surface brightness-dependent axis ratio uncertainties. For this population we infer a mean intrinsic ellipticity E=0.43, and a mean triaxiality T=0.16. This implies that faint Virgo galaxies are best described as a family of thick, nearly oblate spheroids with mean intrinsic axis ratios 1:0.94:0.57. We additionally attempt a study of the intrinsic shapes of Local Group satellites of similar luminosities. For the LG population we infer a slightly larger mean intrinsic ellipticity E=0.51, and the paucity of objects with round apparent shapes translates into more triaxial mean shapes, 1:0.76:0.49. We finally compare the intrinsic shapes of NGVS low-mass galaxies with samples of more massive quiescent systems, and with field, star-forming galaxies of similar luminosities. We find that the intrinsic flattening in this low-luminosity regime is almost independent of the environment in which the galaxy resides--but there is a hint that objects may be slightly rounder in denser environments. The comparable flattening distributions of low-luminosity galaxies that have experienced very different degrees of environmental effects suggests that internal processes are the main drivers of galaxy structure at low masses--with external mechanisms playing a secondary role.