Convex Sets of Finite Population Plans

Let P₁ be a finite population sampling plan and V a collection of subsets of units. The inclusion probabilities for members of V may be calculated. For example, if V comprises all single units and pairs of units we obtain all first and second order inclusion probabilities ᵢ, ₈₉. Another plan P₂ is called equivalent to P₁ with respect to V if the corresponding inclusion probabilities for P₁ are equal to those for P₂. However, P₂ may have fewer samples with positive probability of selection, that is to say smaller "support. " An upper bound is put on the minimum support size of all such P₂. For P₁ simple random sampling, some examples are given for P₂ with small support.