  We prove that C. Loewner's inequality for the torus is satisfied by all
hyperelliptic surfaces X, as well. We first construct the Loewner loops on the
(mildly singular) companion tori, locally isometric to X away from the
Weierstrass points. The loops are then transplanted to X, and surgered to
obtain a Loewner loop on X. In higher genus, we exploit M. Gromov's area
estimates for epsilon-regular metrics on X.
