Can the bounds to system performance in geometrical optics be attained?

The laws of geometrical optics establish that certain imaging properties cannot be realized. For example, it is impossible to image sharply any three-dimensional region with nonunit magnification. Hamilton’s characteristic functions not only offer a simple means to determine whether certain properties are unattainable but have also been used to derive bounds in answer to the more practical question: ”How closely can any particular unattainable ideal be approached?” Basic matters that relate to these bounds and whether they can be attained by realizable systems are considered here. Limits to the task of imaging more than a single plane object are used for illustration. A fundamental constraint has been overlooked in earlier research on this topic and a new bound is derived for the performance of systems in a classic lens design problem, but new questions emerge and remain unanswered.