  Wormholes allowed by the general theory of relativity that are simultaneously
traversable by humanoid travelers are subject to severe constraints from
quantum field theory, particularly the so-called quantum inequalities, here
slightly extended. Moreover, self-collapse of such wormholes can only be
prevented by the use of "exotic matter," which, being rather problematical,
should be used in only minimal quantities. However, making the layer of exotic
matter arbitrarily thin leads to other problems, such as the need for extreme
fine-tuning. This paper discusses a class of wormhole geometries that strike a
balance between reducing the proper distance across the exotic region and the
degree of fine-tuning required to achieve this reduction. Surprisingly, the
degree of fine-tuning appears to be a generic feature of the type of wormhole
discussed. No particular restriction is placed on the throat size, even though
the proper thickness of the exotic region can indeed be quite small. Various
traversability criteria are shown to be met.
