Hadwiger conjecture (combinatorial geometry)

Let $n \ge 1$ and let $K \subset \mathbb{R}^n$ be a convex body, i.e. a bounded closed convex set with nonempty interior. Call a set of the form $sK + v = \{sx + v : x \in K\}$, with $s \in (0,1)$ and $v \in \mathbb{R}^n$, a smaller homothet of $K$, and define

$$h(K) = \min\Big\{ m \in \mathbb{N} : \exists\, s_1,\dots,s_m \in (0,1),\ v_1,\dots,v_m \in \mathbb{R}^n \text{ with } K \subseteq \bigcup_{i=1}^{m} \big(s_i K + v_i\big) \Big\}.$$

Then for every convex body $K \subset \mathbb{R}^n$,

$$h(K) \le 2^n,$$

and $h(K) = 2^n$ holds if and only if $K$ is a parallelepiped, that is, an affine image of the cube $[0,1]^n$ under a nonsingular affine map; in that case the covering may be realized with $s_1 = \cdots = s_{2^n} = 1/2$. Equivalently, for every convex body $K \subset \mathbb{R}^n$ that is not a parallelepiped there exist $m < 2^n$, scalars $s_1,\dots,s_m \in (0,1)$ and vectors $v_1,\dots,v_m \in \mathbb{R}^n$ with $K \subseteq \bigcup_{i=1}^m (s_i K + v_i)$.

In the equivalent illumination formulation, say that a point $p \in \mathbb{R}^n \setminus K$ illuminates a boundary point $x \in \partial K$ if $p$ is separated from $K$ by every hyperplane that supports $K$ at $x$; equivalently, the ray from $p$ through $x$ enters the interior of $K$ immediately beyond $x$. Let $I(K)$ be the least cardinality of a set $P \subset \mathbb{R}^n \setminus K$ such that every $x \in \partial K$ is illuminated by some $p \in P$. Then $I(K) = h(K)$ for every convex body $K$, so the assertion reads: $I(K) \le 2^n$ for every convex body $K \subset \mathbb{R}^n$, with equality precisely when $K$ is a parallelepiped.