The Hadwiger Covering Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A convex body in $\mathbb R^n$ is a compact convex set $K$ with nonempty interior. A smaller positive homothetic copy of $K$ is $t+\lambda K=\{t+\lambda x:x\in K\}$ with $t\in\mathbb R^n$ and $0<\lambda<1$. Different copies may have different translation vectors and scale factors.

Statement recorded in the supplied source.

For every integer $n\geq1$ and every convex body $K\subset\mathbb R^n$, do there exist $m\leq2^n$, vectors $t_1,\ldots,t_m\in\mathbb R^n$, and numbers $0<\lambda_1,\ldots,\lambda_m<1$ such that \[ K\subseteq\bigcup_{j=1}^m(t_j+\lambda_jK)? \] [S2]

Short English statement

Can every convex body be covered by at most two to the dimension smaller translated and positively scaled copies of itself?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1213 (GEO-022). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Andrii Arman, Andriy Bondarenko and Andriy Prymak, On Hadwiger's covering problem in small dimensions, arXiv:2404.00547v1 (2024), Section 1. https://arxiv.org/pdf/2404.00547.