Geometrical foundations of the sampling design with fixed sample size

We study the sampling design with fixed sample size from a geometric point of view. The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. It is possible to study them inside of linear spaces provided with a quadratic and linear metric. We define particular random quantities whose logically possible values are all logically possible samples of a given size. In particular, we define random quantities which are complementary to the Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations. We use the α-criterion of concordance introduced by Gini in order to identify it. We innovatively apply to probability this statistical criterion.