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Search: All articles in the CJM digital archive with keyword convex set

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1. CJM 2012 (vol 65 pp. 1236)

De Bernardi, Carlo Alberto
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)

Dawson, Robert J. MacG.; Moszy\'{n}ska, Maria
\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

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