Convex sets, metric space of
From Encyclopedia of Mathematics
The family of compact convex sets (cf. Convex set) in a Euclidean space endowed with the Hausdorff metric:
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
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