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A problem in combinatorial geometry on the covering of a convex body by figures of a special form, which was put forth by H. Hadwiger in . Let $K$ be a convex body in the $n$-dimensional Euclidean space $\mathbf R^n$, and let $b(K)$ the minimal number of bodies homothetic to $K$ with homothety coefficient $k$, $0<k<1$, that are sufficient to cover $K$. The Hadwiger conjecture consists in the following: Any bounded set $K\subset\mathbf R^n$ satisfies the inequality
$$n+1\leq b(K)\leq2^n.\tag{*}$$
Here the equality $b(K)=2^n$ characterizes a parallelepiped (see ). The Hadwiger conjecture has been proved for $n\leq2$; for $n\geq3$ there are (1988) only partial results. For example, for any $n$-dimensional bounded polyhedron $K\subset\mathbf R^n$ in which any two vertices belong to two distinct parallel supporting hyperplanes to $K$ the inequality \ref{*} holds. Here $b(K)$ coincides with the number of vertices of $K$, but in the set of such polyhedra the equality $b(K)=2^n$ has been verified only for parallelepipeds. This result is connected with the solution of the Erdös problem on the number of points in $\mathbf R^n$ any three of which form a triangle that is not obtuse angled. The Hadwiger conjecture is also connected with covering; decomposition and the illumination problem. For example, the Hadwiger conjecture can be regarded as a generalization of the Borsuk problem on the decomposition of a set into parts of smaller diameter, when $\mathbf R^n$ is replaced by a Minkowski space. For an unbounded set $K\subset\mathbf R^n$ the number $b(K)$ is either equal to $b(K')$, where $K'$ is a convex bounded body of lower dimension, or is $\infty$. For example, for $K\subset\mathbf R^3$ the number $b(K)$ can only take one of the values $1,2,3,4,\infty$ (see ).