scholarly journals Furstenberg boundaries for pairs of groups

2020 ◽  
pp. 1-16
Author(s):  
NICOLAS MONOD
Keyword(s):  
Topological Group ◽  
The Other ◽  
Unitary Representations ◽  
Convex Compact ◽  
Discrete Groups ◽  
Compact Sets ◽  
Link Type ◽  
Relative Notion ◽  
Special Instance ◽  
Convex Compact Sets

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$ . If we consider in addition a subgroup $H<G$ , the relative notion of $(G,H)$ -boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$ . In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$ , namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$ . We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

On minimal convex pairs of convex compact sets

1996 ◽  
Vol 67 (3) ◽  
pp. 226-238 ◽  
Author(s):  
Ryszard Urbański
Keyword(s):  
Convex Compact ◽  
Compact Sets ◽  
Minimal Convex ◽  
Convex Compact Sets

Čebyšev Sets in Hyperspaces over Rn

2009 ◽  
Vol 61 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Robert J. MacG. Dawson and ◽  
Maria Moszyńska
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Linear Structure ◽  
Convex Compact ◽  
Nearest Neighbour ◽  
Compact Sets ◽  
Strictly Convex ◽  
Convex Compact Sets

Abstract. A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over Rn, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

scholarly journals The completion of the space of convex, bounded sets with respect to the Demyanov metric

2013 ◽  
Vol 46 (1) ◽  
Author(s):  
Jerzy Grzybowski ◽  
Andrzej Leśniewski ◽  
Tadeusz Rzeżuchowski
Keyword(s):  
Convex Compact ◽  
Compact Sets ◽  
Finite Dimensional ◽  
The Family ◽  
Bounded Sets ◽  
Convex Compact Sets

AbstractThe Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in ℝ

scholarly journals Convex-compact sets and Banach discs

2009 ◽  
Vol 59 (3) ◽  
pp. 773-780
Author(s):  
I. Monterde ◽  
V. Montesinos
Keyword(s):  
Convex Compact ◽  
Compact Sets ◽  
Convex Compact Sets
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