Before revision: We prove a no-dimensional version of Carath\'edory's theorem: given an n-element set P\subset \mathbb{R ^d, a point a \in \mathrm{conv \conv P, and an integer r\le d, r \le n, there is a subset Q\subset P of r elements such that the distance between a and \mathrm{conv \conv Q is less then \mathrm{diam\diam P/\sqrt {2r}.
After revision: We prove a no-dimensional version of Carath\'edory's theorem: given an n-element set P\subset \mathbb{R ^d, a point a \in \mathrm{conv \conv P, and an integer r\le d, r \le n, there is a subset Q\subset P of r elements such that the distance between a and \mathrm{conv \conv Q is less than\diam P/\sqrt {2r}.