It is impractical to directly constrain the elastic constants of a transversely isotropic (TI) medium using travel-time data. In contrast, as we show in this paper, the elastic constants can be determined straightforwardly by inversion of body-wave waveform data. We invert the horizontal components of observed seismic waveforms of S and ScS phases (as well as any other phases arriving in the time window) to determine the radial profile of TI shear wave velocity in the lowermost mantle beneath the Pacific. We find that the radial (SV) component is faster than the transverse (SH) component in the depth range from about 200-400 km above the core-mantle boundary (CMB). The major mineralogical components above the D″ discontinuity in this depth range are Mg-perovskite (pv) and ferropericlase (fp). The observed anisotropy can be interpreted as due to lattice preferred orientation (LPO) of either pv, fp, or both in the lowermost mantle induced by vertical flow due to thermal buoyancy, which might be related to the origin of the Hawaiian hotspot (although other possibilities such as a chemically distinct layer, LPO of post-perovskite (ppv), or LPO in counter-flow in and around a chemically dense pile cannot be excluded). We show that resolution of the velocity of SV shear waves very close to the CMB is inherently limited due to the boundary condition of zero tangential traction at the CMB; shear wave splitting studies thus cannot be used to investigate the anisotropy of the lowermost mantle.