Theorems of Carathéodory, Helly, and Tverberg Without Dimension

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤ n, there is a subset Q⊂ P of r elements such that the distance between a and convQ is less than diamP/2r. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of R^d. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ R^d which is closer than 1/k to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.