Convex and discrete geometry Convex geometric analysis and geometric inequalities Conjecture

Hadwiger covering conjecture

Statement

Let $n \ge 1$ , and let $K \subseteq \R^n$ be a convex body. Then there are $\lambda \in (0, 1)$ and points $x_1, \ldots, x_{2^n} \in \R^n$ with $K \subseteq \bigcup_{i=1}^{2^n} (x_i + \lambda K),$ that is, $K$ can be covered by $2^n$ smaller homothetic copies of itself. Moreover, $2^n$ copies are needed only when $K$ is a parallelotope.

Remarks

The conjecture is open for $n \ge 3$ ; Levi proved the case $n = 2$ . The cube $[0, 1]^n$ shows that $2^n$ cannot be lowered. A translate of $\lambda [0, 1]^n$ with $\lambda < 1$ has all side lengths below $1$ , so it contains at most one of the $2^n$ vertices. General upper bounds of exponential order, about $4^n$ times a polynomial factor in $n$ , are known. Boltyanski showed that the conjecture is equivalent to a statement about illuminating the boundary of $K$ by $2^n$ directions.

Reading path

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Illumination problem

Tag	`SVP75YY386`
URI	`m2://SVP75YY386`
Kind	conjecture
Writing	W0 stub
MSC	52C17, 52A20
Status	This copy of the site has no build data. Run `m2 status SVP75YY386`.
About	Convex body
Uses	Convex body, Homothety
Same as	Wikidata Q5638115 Wikipedia Hadwiger conjecture (combinatorial geometry)

Hadwiger covering conjecture

Statement

Remarks

Reading path

Where this leads

Further reading

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