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1. CJM 2012 (vol 65 pp. 1236)
Higher Connectedness Properties of Support Points and Functionals of Convex Sets We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\mathrm{AR(}\sigma$-$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$.
Keywords:convex set, support point, support functional, absolute retract, Leray-Schauder continuation principle Categories:46A55, 46B99, 52A07 |
2. CJM 2009 (vol 61 pp. 299)
\v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$ A set in a metric space is called a \emph{\v{C}eby\v{s}ev 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
\emph{\v{C}eby\v{s}ev relative to} another set if every point in the
second set has a unique ``nearest neighbour'' in the first. We are
interested in \v{C}eby\v{s}ev sets in some hyperspaces over $\R$,
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 \v{C}eby\v{s}ev and relatively
\v{C}eby\v{s}ev sets in various hyperspaces. In particular, we show
that certain nested families of sets are \v{C}eby\v{s}ev. As these
families are characterized purely in terms of containment, without
reference to the semi-linear structure of the underlying metric space,
their properties differ markedly from those of known \v{C}eby\v{s}ev
sets.
Keywords:convex body, strictly convex set, \v{C}eby\v{s}ev set, relative \v{C}eby\v{s}ev set, nested family, strongly nested family, family of translates Categories:41A52, 52A20 |