Theorems of Carathéodory, Helly, and Tverberg Without Dimension

Abstract

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 $$a \in {{\,\mathrm{{\texttt {conv}}}\,}}P$$ , and an integer $$r \le n$$ , there is a subset $$Q\subset P$$ of r elements such that the distance between a and $${{\,\mathrm{{\texttt {conv}}}\,}}Q$$ is less than $${{\,\mathrm{{\texttt {diam}}}\,}}P/\sqrt{2r}$$ . In an analoguos no-dimension Helly theorem a finite family $$\mathcal {F}$$ of convex bodies is given, all of them are contained in the Euclidean unit ball of $$\mathbb {R}^d$$ . If $$k\le d$$ , $$|\mathcal {F}|\ge k$$ , and every k-element subfamily of $$\mathcal {F}$$ is intersecting, then there is a point $$q \in \mathbb {R}^d$$ which is closer than $$1/\sqrt{k}$$ to every set in $$\mathcal {F}$$ . This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.