Theorems of Carath'eodory, Helly, and Tverberg without dimension

We prove a no-dimensional version of Carath'edory's theorem: given an n-element set Psubset Red, a point a ∈ conv P, and an integer r≤ d, r ≤ n, there is a subset Qsubset P of r elements such that the distance between a and conv Q is less than diam P/√ 2r. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.