Monotonicity properties of keff with shape change and with nesting

It was found that, contrary to expectations based on physical intuition, keff can both increase and decrease when changing the shape of an initially regular critical system, while preserving its volume. Physical intuition would only allow for a decrease of keff when the surface/volume ratio increases. The unexpected behaviour of increasing keff was found through numerical investigation. For a convincing demonstration of the possibility of the non-monotonic behaviour, a simple geometrical proof was constructed. This latter proof, in turn, is based on the assumption that keff can only increase (or stay constant) in the case of nesting, i.e. when adding extra volume to a system. Since we found no formal proof of the nesting theorem for the general case, we close the paper by a simple formal proof of the monotonic behaviour of keff by nesting.