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Difference between revisions of "Convex sets, metric space of"

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The family of compact convex sets (cf. [[Convex set|Convex set]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026360/c0263601.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026360/c0263602.png" /> endowed with the Hausdorff metric:
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The family of compact convex sets (cf. [[Convex set]]) $F$ in a Euclidean space $E^n$ endowed with the [[Hausdorff metric]]:
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026360/c0263603.png" /></td> </tr></table>
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\rho(F_1,F_2) = \sup_{x\in F_1,\,y\in F_2} \{\rho(x,F_2), \rho(y,F_1) \}
 
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$$
This space is boundedly compact (cf. [[Blaschke selection theorem|Blaschke selection theorem]]). For analogues of metric spaces of convex sets (other metrizations, non-compact sets, classes of sets, other initial spaces) see [[#References|[1]]].
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This space is [[Boundedly-compact set|boundedly compact]] (cf. [[Blaschke selection theorem]]). For analogues of metric spaces of convex sets (other metrizations, non-compact sets, classes of sets, other initial spaces) see [[#References|[1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Grünbaum,  "Measures of symmetry for convex sets" , ''Proc. Symp. Pure Math.'' , '''7 (Convexity)''' , Amer. Math. Soc.  (1963)  pp. 233–270</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B. Grünbaum,  "Measures of symmetry for convex sets" , ''Proc. Symp. Pure Math.'' , '''7 (Convexity)''' , Amer. Math. Soc.  (1963)  pp. 233–270</TD></TR>
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,  "Approximation of convex bodies"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 131–162</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Hadwiger,  "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer  (1957)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Schneider,  "Valuations on convex bodies"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 170–247</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,  "Approximation of convex bodies"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 131–162</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Hadwiger,  "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer  (1957)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Schneider,  "Valuations on convex bodies"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 170–247</TD></TR>
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</table>
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[[Category:Topological spaces with richer structure]]

Latest revision as of 17:00, 17 October 2014

The family of compact convex sets (cf. Convex set) $F$ in a Euclidean space $E^n$ endowed with the Hausdorff metric: $$ \rho(F_1,F_2) = \sup_{x\in F_1,\,y\in F_2} \{\rho(x,F_2), \rho(y,F_1) \} $$ This space is boundedly compact (cf. Blaschke selection theorem). For analogues of metric spaces of convex sets (other metrizations, non-compact sets, classes of sets, other initial spaces) see [1].

References

[1] B. Grünbaum, "Measures of symmetry for convex sets" , Proc. Symp. Pure Math. , 7 (Convexity) , Amer. Math. Soc. (1963) pp. 233–270


Comments

Metric spaces of convex sets (in particular the metrization by the symmetric difference metric) play a basic role in the foundations of analysis in convex geometry. New important results in convex geometry are given in [a1], [a3]; [a2] gives a general and axiomatic approach.

References

[a1] P.M. Gruber, "Approximation of convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 131–162
[a2] H. Hadwiger, "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer (1957)
[a3] R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 170–247
How to Cite This Entry:
Convex sets, metric space of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_sets,_metric_space_of&oldid=17279
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article