  We are concerned with the dependence of the lowest positive eigenvalue of the
Dirac operator on the geometry of rectangles, subject to infinite-mass boundary
conditions. We conjecture that the square is a global minimiser both under the
area or perimeter constraints. Contrary to well-known non-relativistic
analogues, we show that the present spectral problem does not admit explicit
solutions. We prove partial optimisation results based on a variational
reformulation and newly established lower and upper bounds to the Dirac
eigenvalue. We also propose an alternative approach based on symmetries of
rectangles and a non-convex minimisation problem; this implies a sufficient
condition formulated in terms of a symmetry of the minimiser which guarantees
the conjectured results.
