Remarks on Bárányʹs theorem and affine selections

Using the ‘multiplied’ version of Hellyʹs theorem given by Bárány (Discrete Math. 40 (1982) 141–152) we generalize some selection and separation results obtained in Behrends and Nikodem (Studia Math. 116 (1) (1995) 43–48), Nikodem and Wa̧sowicz (Aequationes Math. 49 (1995) 160–164) and Wa̧sowicz (J. Appl. Anal. 1 (2) (1995) 173–179). In particular, it is shown that if three set-valued functions Φ1,Φ2,Φ3:I→cc(R) satisfy the condition Φi(tx+(1−t)y)∩[tΦj(x)+(1−t)Φk(y)]≠∅ for all x,y∈I,t∈[0,1] and every permutation (i,j,k) of the set {1,2,3} then at least one of them has an affine selection.