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 Rd. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ Rd 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.
Bibliographical noteFunding Information:
K.A. was supported by ERC StG 716424-CASe and ISF Grant 1050/16. I.B. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants K 111827 and K 116769, and by ERC-AdG 321104. N.M. was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01). T.T. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants NK 112735 and K 120697.
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- Carathéodory theorem
- Convex approximation
- Tverberg theorem