Shape optimization for a rectangularly constrained small loop antenna

The theoretical optimal shape of a small loop antenna within a rectangular area restriction is derived. Approaches with cut-off corners and rounded corners are proposed and optimal parameters (such as cut-off radius) are calculated. By means of the calculus of variation, a proof is given that the rounded-corner approach is indeed the optimal shape in terms of antenna efficiency if a simple model is assumed; the result is valid for any convex shape constraint.