Theorems of Carathéodory, Helly, and Tverberg without dimension
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We prove a no-dimensional version of Carathédory's theorem: given an n-element set P⊂R^d, a point a ∈conv P, and an integer r< d, r < n, there is a subset Q⊂ P of r elements such that the distance between a and conv Q is less then 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.
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