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

中文翻译：

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Carathéodory、Helly 和 Tverberg 的无量纲定理

我们开始研究凸性经典定理的无量纲版本。一个例子是无量纲的 Carathéodory 定理：给定欧几里得空间中的 n 元素集 P，点 $$a \in {{\,\mathrm{{\texttt {conv}}}\,}}P$$ a ∈ conv P ，和一个整数 $$r \le n$$ r ≤ n ，有一个子集 $$Q\subset P$$ Q ⊂ P 的 r 个元素使得 a 和 $${{\, \mathrm{{\texttt {conv}}}\,}}Q$$ conv Q 小于 $${{\,\mathrm{{\texttt {diam}}}\,}}P/\sqrt{2r }$$ 直径 P / 2 r 。在类似的无维海利定理中，给出了凸体的有限族 $$\mathcal {F}$$ F，它们都包含在 $$\mathbb {R}^d$$ R 的欧几里德单位球中d。如果 $$k\le d$$ k ≤ d ， $$|\mathcal {F}|\ge k$$ | F | ≥ k ，并且 $$\mathcal {F}$$ F 的每个 k 元素子族是相交的，那么有一个点 $$q \in \mathbb {R}^d$$ q ∈ R d 比 $$1/\sqrt{k}$$ 1 / k 更接近 $$\mathcal {F }$$ F。这个结果有几个丰富多彩的部分结果。还建立了 Tverberg 定理的类似版本及其一些扩展。