STUDIA MATHEMATICA 158 (1) (2003) On the number of minimal pairs of compact convex sets that are not translates of one another by J. Grzybowski and R. Urbański (Poznań) Abstract. Let [A, B] be the family of pairs of compact convex sets equivalent to (A, B). We prove that the cardinality of the set of minimal pairs in [A, B] that are not translates of one another is either 1 or greater than ℵ 0 . Let X =(X, τ ) be a topological vector space over the field R. Let K(X ) be the family of all nonempty compact convex subsets of X . For any A, B ⊂ X the Minkowski sum is defined by A + B = {a + b | a ∈ A and b ∈ B}. For (A, B), (C, D) ∈K 2 (X ), let (A, B) ∼ (C, D) if and only if A+D = B+C . Let [A, B] be the equivalence class of (A, B) in K 2 (X )/∼. For (A, B), (C, D) ∈ K 2 (X ) let (A, B) ≤ (C, D) if and only if (A, B) ∼ (C, D), A ⊂ C and B ⊂ D. Let m[A, B] be the family of all elements of [A, B] that are minimal with respect to the ordering ≤. Let A ∨ B be the convex hull of A ∪ B. For A,B,C ∈K 2 (X ), we have the Pinsker formula A∨B +C =(A+C ) ∨(B +C ). Minimal pairs of compact convex sets play an important role in quasi- differential calculus [5]–[7]. Minimal pairs were studied in numerous papers ([1]–[4], [8]–[14], [17], and others). Let (A, B) ∈K 2 (X ) and n A,B be the number of minimal pairs in m[A, B] that are not translates of one another. If X = R 1 or R 2 then n A,B is always 1 ([8], [15]). In [13], there is an example of A, B ∈K(R 3 ) such that n A,B is the continuum. In December 2000, Professor S. Rolewicz posed the problem whether n A,B can be finite and greater than 1. The following theorem implies a negative answer to this problem. Theorem. Let (A 1 ,B 1 ), (A 2 ,B 2 ) be two equivalent minimal pairs of compact convex sets such that (A 2 ,B 2 ) is not a translate of (A 1 ,B 1 ). Then there exists an uncountable family (A λ ,B λ ), λ ∈ Λ, of minimal pairs that are equivalent to (A 1 ,B 1 ) and no (A λ ,B λ ) is a translate of (A μ ,B μ ), λ = μ. 2000 Mathematics Subject Classification : 52A07, 26A27. Key words and phrases : convex analysis, minimal pairs of compact convex sets. [59]