On the hyperspace of max-min convex compact sets

Abstract

A subset $A$ of $\mathbb R^n$ is said to be max-min convex if, for any $x,y\in A$ and any $t\in \mathbb R$, we have $x\oplus t\otimes y\in A$ (here $\oplus$ stands for the coordinatewise maximum of two elements in $\mathbb R^n$ and $t\otimes (y_1,\dots,y_n)=(\min\{t,y_1\},\dots, \min\{t,y_n\})$). It is proved that the hyperspace of compact max-min convex sets in the Euclidean space $\mathbb R^n$, $n\ge2$, is homeomorphic to the punctured Hilbert cube. This is a counterpart of the result by Nadler, Quinn and Stavrokas proved for the hyperspace of compact convex sets. We also investigate the maps of the hyperspaces of compact max-min convex sets induced by the projection maps of Euclidean spaces. It is proved that this map is a Hilbert cube manifold bundle.

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Article Information

Title	On the hyperspace of max-min convex compact sets
Source	Methods Funct. Anal. Topology, Vol. 15 (2009), no. 4, 322-332
MathSciNet	MR2603838
Copyright	The Author(s) 2009 (CC BY-SA)

Authors Information

L. E. Bazylevych
Institute of Applied Mathematics and Fundamental Sciences, National University "Lviv Polytechnica", 5 Mytropolyta Andreya Str., Lviv, 79013, Ukraine 

Citation Example

L. E. Bazylevych, On the hyperspace of max-min convex compact sets, Methods Funct. Anal. Topology 15 (2009), no. 4, 322-332.

BibTex

@article {MFAT442,
    AUTHOR = {Bazylevych, L. E.},
     TITLE = {On the  hyperspace of max-min convex compact sets},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {15},
      YEAR = {2009},
    NUMBER = {4},
     PAGES = {322-332},
      ISSN = {1029-3531},
  MRNUMBER = {MR2603838},
       URL = {http://mfat.imath.kiev.ua/article/?id=442},
}