L. E. Bazylevych

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On the hyperspace of max-min convex compact sets

L. E. Bazylevych

↓ Abstract   |   Article (.pdf)

MFAT 15 (2009), no. 4, 322-332


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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