View source for 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|decomposition]] into not more than $n+1$ subsets with diameters smaller than $d$? The problem was formulated by K. Borsuk [[#References|[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 $\mathbf R^n$ [[#References|[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 $F$ [[#References|[3]]]. This fact cannot be directly generalized to the case $n>2$. The Borsuk problem is closely related to the [[Illumination problem|illumination problem]] and to the [[Hadwiger hypothesis|Hadwiger hypothesis]], which is a generalization of the Borsuk problem in which $\mathbf R^n$ is replaced by a finite-dimensional normed space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Borsuk,   "Drei Sätze über die $n$-dimensionale euklidische Sphäre"  ''Fund. Math.'' , '''20'''  (1933)  pp. 177–190</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.G. Boltyanskii,   "On decomposition of plane figures in parts of least diameter"  ''Colloq. Math.'' , '''21''' :  2  (1967)  pp. 253–263  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Boltyanskii,   I.Ts. Gokhberg,   "Sätze und Probleme der Kombinatorische Geometrie" , Deutsch. Verlag Wissenschaft.  (1972)  (Translated from Russian)</TD></TR></table>