A generalization of carathéodory's theorem

The well-known Carath&odory’s theorem says that, given a set VG$” and a point a E conv V (the convex hull of V), there exists a subset A c V such that IAl s n + 1 and a E conv A. This simple theorem has many applications and generalizations (see, for instance, [4, 9, 1OJ). The aim of this paper is to give a new generalization of Carathiodory’s theorem and to present some consequences of thig generalization. The paper is organized as follows. The second section contains the main theorems. The third section is about a generalization of Helly’s theorem. The fourth section deals with systems of linear inequalities and simple polytcjpes. The next section answers a question of Boros and Ftiedi [2]. In the last se&an we apply our results to convex functions.