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A quantitative program for Hadwiger’s covering conjecture

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Abstract

In 1957, Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its interior. Up to now, this conjecture is still open for all n ⩾ 3. In 1933, Borsuk made a conjecture that every n-dimensional bounded set can be divided into n + 1 subsets of smaller diameters. Up to now, this conjecture is open for 4 ⩽ n ⩽ 297. In this article we encode the two conjectures into continuous functions defined on the spaces of convex bodies, propose a four-step program to attack them, and obtain some partial results.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Chuanming Zong

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  1. Chuanming Zong
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Correspondence to Chuanming Zong.

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Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

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Zong, C. A quantitative program for Hadwiger’s covering conjecture. Sci. China Math. 53, 2551–2560 (2010). https://doi.org/10.1007/s11425-010-4087-3

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  • DOI: https://doi.org/10.1007/s11425-010-4087-3

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