A quantitative program for Hadwiger’s covering conjecture

By Chuanming Zong. 2009

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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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