Convex sets, metric space of

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