Characterizations of strictly convex sets by the uniqueness of support points

Optimization ◽  
2018 ◽  
Vol 68 (7) ◽  
pp. 1321-1335 ◽  
Author(s):  
Truong Xuan Duc Ha ◽  
Johannes Jahn
Keyword(s):  
Convex Sets ◽  
Strictly Convex ◽  
Support Points

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.

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Convex Subset ◽  
Convex Sets ◽  
Lower Semicontinuous ◽  
Nonempty Closed Convex Subset ◽  
Common Point ◽  
Support Points ◽  
Support Theorems

AbstractThe Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

scholarly journals Rigidity of complex convex divisible sets

2018 ◽  
Vol 10 (04) ◽  
pp. 817-851
Author(s):  
Andrew M. Zimmer
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Discrete Group ◽  
Convex Sets ◽  
Convex Set ◽  
Real Projective Space ◽  
Open Convex ◽  
Strictly Convex ◽  
Complex Projective

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

Weak* support points of convex sets inE*

1964 ◽  
Vol 2 (3) ◽  
pp. 177-182 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Convex Sets ◽  
Weak Support ◽  
Support Points

Higher Connectedness Properties of Support Points and Functionals of Convex Sets

2013 ◽  
Vol 65 (6) ◽  
pp. 1236-1254
Author(s):  
Carlo Alberto De Bernardi
Keyword(s):  
Convex Sets ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Infinite Dimensional ◽  
Support Points ◽  
Bounded Set ◽  
Infinite Dimensional Banach Space ◽  
Proper Convex ◽  
Convex Lower Semicontinuous Function

AbstractWe 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 AR(σ-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 lower semicontinuous function on X.

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