In this almost 30-pages paper the authors study questions (tests) about consistence of observations of price vectors of a good and output of firms (data sets) with the Cournot hypothesis. At first they show, that any data set is consistent with the Cournot hypothesis. But there are restrictions for the observations in their model because of the common assumption of convexity for the cost functions. So it follows a discussion and a test on Cournot rationalizability of data sets (Cournot equilibrium) with respect to that convexity. That test is connected with a linear optimization problem and is -- as the authors write -- easy to implement and useful. Then, by special families of convex cost functions, a connection to compact and convex sublattices of the Euclidean space is constructed and algorithms with not more steps than the number of observations, are given looking for the supremum and infimum (marginal costs) of the considered sublattices. Two chapters are devoted to questions of market structure such as f.i. symmetry of market interaction, and things which may affect the firm's cost functions such as f.i. collusive interaction of firms. In order to show the possibility to use the new algorithms to real-world problems, the authors apply them to the oil industry. A lot of hints and examples supplement the paper.