One of the shortest and the most elegant proofs of the well known Tverberg theorem, due to Karanbir Sarkaria, relies on a generalization of Carathéodory's theorem found by Imre Bárány. Our objective is to set up the stage for studying other Carathéodory--Bárány type results which should open a possibility for similar proofs of related Tverberg type statements. Among the Tverberg type results which motivate this study are colored Tverberg theorems, [2], [6], [15], [20] and more recent results related to the Tverberg--Vrećica conjecture, [17], [18].