Borsuk problem

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One of the fundamental problems in combinatorial geometry: Is it possible, for any bounded set of diameter $d>0$ in an $n$-dimensional Euclidean space, to make a decomposition into not more than $n+1$ subsets with diameters smaller than $d$? The problem was formulated by K. Borsuk [1] who noted that it was not possible to subdivide an $n$-dimensional simplex and an $n$-dimensional sphere in $\mathbf R^n$ into $n$ parts of a smaller diameter. The problem has a positive solution for $n=2,3$, but only partial results have been obtained for $n>3$. Thus, for instance, the problem has been positively solved for any bounded smooth convex set in $\mathbf R^n$ [2]. It has been proved that the solution of Borsuk's problem can be reduced to the case of sets of constant width. If $a(F)$ is the smallest number of parts of a diameter smaller than $d$ into which a set $F\subset\mathbf R^n$ can be subdivided, then the equality $a(F)=3$ is valid for a figure $F\subset\mathbf R^2$ of diameter $d$ if and only if $\mathbf R^2$ contains a unique figure of constant width $d$ containing $F$ [3]. This fact cannot be directly generalized to the case $n>2$. The Borsuk problem is closely related to the illumination problem and to the Hadwiger hypothesis, which is a generalization of the Borsuk problem in which $\mathbf R^n$ is replaced by a finite-dimensional normed space.

References

 [1] K. Borsuk, "Drei Sätze über die $n$-dimensionale euklidische Sphäre" Fund. Math. , 20 (1933) pp. 177–190 [2] B. Grünbaum, "Borsuk's problem and related questions" V.L. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 271–284 [3] V.G. Boltyanskii, "On decomposition of plane figures in parts of least diameter" Colloq. Math. , 21 : 2 (1967) pp. 253–263 (In Russian)