Hadwiger hypothesis

Hadwiger conjecture

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 [1]. 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 [1]). 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 [2]).

References

[1]	H. Hadwiger, "Ueber Treffanzahlen bei translationsgleichen Eikörpern" Arch. Math. (Basel) , 8 (1957) pp. 212–213
[2]	V.G. Boltyanskii, P.S. Soltan, "The combinatorial geometry of various classes of convex sets" , Kishinev (1978) (In Russian)

Comments

For bounded centrally-symmetric bodies $K\subset\mathbf R^3$ Hadwiger's conjecture holds, see [a1].

See also Geometry of numbers and the standard work [a4].

References

[a1]	M. Lassak, "Solution of Hadwiger's covering problem for centrally symmetric convex bodies in $E^3$" J. London Math. Soc. (2) , 30 (1984) pp. 501–511
[a2]	L. Danzer, B. Grünbaum, V.L. Klee, "Helly's theorem and its relatives" V.L. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 101–180
[a3]	H. Hadwiger, H. Debrunner, "Kombinatorische Geometrie in der Ebene" L'Enseign. Math. , 2 (1959)
[a4]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)