This work concerns the identification of the structure of a genetic network model from measurements of gene product concentrations and synthesis rates. In earlier work, we developed a data preprocessing algorithm that is able to reject many hypotheses on the network structure by testing certain monotonicity properties for a wide family of network models. Here we develop a geometric interpretation of the method. Then, for a relevant subclass of genetic network models, we extend our approach to the combined testing of monotonicity and convexity-like properties associated with the network structures. The theoretical aspects and practical performance of the enhanced methods are illustrated by way of numerical results.