scholarly journals Characterisation of drop and weak drop properties for closed bounded convex sets

1991 ◽  
Vol 43 (3) ◽  
pp. 377-385 ◽  
Author(s):  
J.R. Giles ◽  
Denka N. Kutzarova
Keyword(s):  
Convex Sets ◽  
Drop Theorem ◽  
Bounded Convex

Modifying the concept underlying Daneš' drop theorem, Rolewicz introduced the notion of the drop property of a norm which was later generalised to the weak drop property of a norm. Kutzarova extended the discussion to consider the drop property for closed bounded convex sets. Here we characterise the drop and weak drop properties for such sets by upper semi-continuous and compact valued subdifferential mappings.

scholarly journals About an Erdős–Grünbaum Conjecture Concerning Piercing of Non-bounded Convex Sets

2015 ◽  
Vol 53 (4) ◽  
pp. 941-950
Author(s):  
Amanda Montejano ◽  
Luis Montejano ◽  
Edgardo Roldán-Pensado ◽  
Pablo Soberón
Keyword(s):  
Convex Sets ◽  
Bounded Convex

On the Vector Sum of Two Convex Sets in Space

1991 ◽  
Vol 43 (2) ◽  
pp. 347-355 ◽  
Author(s):  
Steven G. Krantz ◽  
Harold R. Parks
Keyword(s):  
Convex Sets ◽  
Analytic Boundary ◽  
Real Analytic ◽  
Vector Sum ◽  
Bounded Convex

In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C20/3 (this result is sharp). When A and B have C∞ boundaries, then the smoothness of the sum set breaks down at the level C5 (see [KIS2] for the various pathologies that arise).

A classification for non-linearly bounded convex sets

1993 ◽  
Vol 61 (6) ◽  
pp. 576-583
Author(s):  
J. Bair ◽  
J. L. Valein
Keyword(s):  
Convex Sets ◽  
Bounded Convex

Order cancellation law in the family of bounded convex sets

2019 ◽  
Vol 77 (2) ◽  
pp. 289-300
Author(s):  
J. Grzybowski ◽  
R. Urbański
Keyword(s):  
Convex Sets ◽  
The Family ◽  
Bounded Convex

scholarly journals Ul’yanov-type inequality for bounded convex sets in Rd

2008 ◽  
Vol 151 (1) ◽  
pp. 60-85 ◽  
Author(s):  
Z. Ditzian ◽  
A. Prymak
Keyword(s):  
Convex Sets ◽  
Type Inequality ◽  
Bounded Convex

scholarly journals The completeness of a normed space is equivalent to the homogeneity of its space of closed bounded convex sets

2013 ◽  
Vol 5 (1) ◽  
pp. 44-46
Author(s):  
I. Hetman
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Infinite Dimensional ◽  
Dimensional Normed Space ◽  
Bounded Convex ◽  
Convex Subsets

We prove that an infinite-dimensional normed space $X$ is complete if and only if the space $\mathrm{BConv}_H(X)$ of all non-empty bounded closed convex subsets of $X$ is topologically homogeneous.

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
Author(s):  
Lixin Cheng ◽  
Zhenghua Luo ◽  
Yu Zhou
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Lower Semicontinuous ◽  
Weakly Compact ◽  
Bounded Set ◽  
Fréchet Differentiable ◽  
Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

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