# Difference between revisions of "Borsuk problem"

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− | One of the fundamental problems in combinatorial geometry: Is it possible, for any bounded set of diameter | + | {{TEX|done}} |

+ | 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==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Borsuk, "Drei Sätze über die | + | <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> |

## Latest revision as of 09:50, 19 April 2014

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) |

#### Comments

#### References

[a1] | V.G. Boltyanskii, I.Ts. Gokhberg, "Sätze und Probleme der Kombinatorische Geometrie" , Deutsch. Verlag Wissenschaft. (1972) (Translated from Russian) |

**How to Cite This Entry:**

Borsuk problem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Borsuk_problem&oldid=18669