Remarks on Bárány's theorem and affine selections

Abstract

Using the ‘multiplied’ version of Helly's 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 Φ 1,Φ 2,Φ 3:I→ cc(R) satisfy the condition Φ i(tx+(1−t)y)∩[tΦ j(x)+(1−t)Φ k(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.