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One of the fundamental problems in combinatorial geometry: Is it possible, for any bounded set of diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172001.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172002.png" />-dimensional Euclidean space, to make a [[Decomposition|decomposition]] into not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172003.png" /> subsets with diameters smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172004.png" />? The problem was formulated by K. Borsuk [[#References|[1]]] who noted that it was not possible to subdivide an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172005.png" />-dimensional simplex and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172006.png" />-dimensional sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172007.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172008.png" /> parts of a smaller diameter. The problem has a positive solution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b0172009.png" />, but only partial results have been obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720010.png" />. Thus, for instance, the problem has been positively solved for any bounded smooth convex set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720011.png" /> [[#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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720012.png" /> is the smallest number of parts of a diameter smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720013.png" /> into which a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720014.png" /> can be subdivided, then the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720015.png" /> is valid for a figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720016.png" /> of diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720017.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720018.png" /> contains a unique figure of constant width <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720019.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720020.png" /> [[#References|[3]]]. This fact cannot be directly generalized to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720021.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720022.png" /> is replaced by a finite-dimensional normed space.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Borsuk,  "Drei Sätze über die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017200/b01720023.png" />-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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<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
This article was adapted from an original article by P.S. Soltan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article