Convex Sets, Cantor Sets and a Midpoint Property

1976 ◽  
Vol 19 (4) ◽  
pp. 467-471 ◽  
Author(s):  
Harold Reiter
Keyword(s):  
Final Section ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Sufficient Conditions ◽  
Convex Polytope ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Compact Convex Set ◽  
Necessary And Sufficient

It is well known that every point of the closed unit interval I can be expressed as the midpoint of two points of the Cantor ternary set D. See [2, p. 549] and [3, p. 105]. Regarding J as a one dimensional compact convex set, it seems natural to try to generalize the above result to higher dimensional convex sets. We prove in section 3 that every convex polytope K in Euclidean space Rd contains a topological copy C of D such that each point of K is expressible as a midpoint of two points of C. Also, we give necessary and sufficient conditions on a planar compact convex set for it to contain a copy of D with the midpoint property above. In the final section we prove a result on minimal midpoint sets.

A geometric characterization concerning compact, convex sets

1991 ◽  
Vol 109 (2) ◽  
pp. 351-361 ◽  
Author(s):  
Christopher J. Mulvey ◽  
Joan Wick Pelletier
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Necessary And Sufficient Condition ◽  
Categorical Logic ◽  
Compact Convex Set ◽  
Necessary And Sufficient ◽  
Formal Statement

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

Rates of convergence for random approximations of convex sets

1996 ◽  
Vol 28 (02) ◽  
pp. 384-393 ◽  
Author(s):  
Lutz Dümbgen ◽  
Günther Walther
Keyword(s):  
Convex Subset ◽  
Hausdorff Distance ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Almost Sure Convergence ◽  
Rates Of Convergence ◽  
Random Sets ◽  
Compact Convex Set

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (02) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

scholarly journals Proximinal subspaces ofA(K)of finite codimension

2003 ◽  
Vol 2003 (39) ◽  
pp. 2501-2505
Author(s):  
T. S. S. R. K. Rao
Keyword(s):  
Compact Subset ◽  
Convex Set ◽  
Compact Convex ◽  
Continuous Functions ◽  
Finite Codimension ◽  
Sufficient Condition ◽  
Choquet Simplex ◽  
Necessary And Sufficient Condition ◽  
Compact Convex Set ◽  
Necessary And Sufficient

We study an analogue of Garkavi's result on proximinal subspaces ofC(X)of finite codimension in the context of the spaceA(K)of affine continuous functions on a compact convex setK. We give an example to show that a simple-minded analogue of Garkavi's result fails for these spaces. WhenKis a metrizable Choquet simplex, we give a necessary and sufficient condition for a boundary measure to attain its norm onA(K). We also exhibit proximinal subspaces of finite codimension ofA(K)when the measures are supported on a compact subset of the extreme boundary.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (2) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

scholarly journals Controllability of networked higher-dimensional systems with one-dimensional communication

2017 ◽  
Vol 375 (2088) ◽  
pp. 20160215 ◽  
Author(s):  
Lin Wang ◽  
Xiaofan Wang ◽  
Guanrong Chen
Keyword(s):  
Network Topology ◽  
Linear Time ◽  
Sufficient Conditions ◽  
External Control ◽  
One Dimensional ◽  
Networked System ◽  
Linear Time Invariant ◽  
Higher Dimensional ◽  
Necessary And Sufficient ◽  
Time Invariant

In this paper, the state controllability of networked higher-dimensional linear time-invariant dynamical systems is considered, where communications are performed through one-dimensional connections. The influences on the controllability of such a networked system are investigated, which come from a combination of network topology, node-system dynamics, external control inputs and inner interactions. Particularly, necessary and sufficient conditions are presented for the controllability of the network with a general topology, as well as for some special settings such as cycles and chains, which show that the observability of the node system is necessary in general and the controllability of the node system is necessary for chains but not necessary for cycles. Moreover, two examples are constructed to illustrate that uncontrollable node systems can be assembled to a controllable networked system, while controllable node systems may lead to uncontrollable systems even for the cycle topology. This article is part of the themed issue ‘Horizons of cybernetical physics’.

Weakly prime compact convex sets and uniform algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 225-232 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Uniform Algebra ◽  
Reduction Theory ◽  
Natural Analogue ◽  
Uniform Algebras ◽  
Compact Convex Set ◽  
Analytic Sets

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

scholarly journals Additive selections and the stability of the Cauchy functional equation

2003 ◽  
Vol 44 (3) ◽  
pp. 323-337 ◽  
Author(s):  
Roman Badora ◽  
Roman Ger ◽  
Zsolt Páles
Keyword(s):  
Convex Set ◽  
Compact Convex ◽  
Necessary And Sufficient Condition ◽  
Cauchy Functional Equation ◽  
Amenable Semigroup ◽  
Weakly Compact ◽  
Compact Convex Set ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Selection Of

AbstractThe main result of this paper offers a necessary and sufficient condition for the existence of an additive selection of a weakly compact convex set-valued map defined on an amenable semigroup. As an application, we obtain characterisations of the solutions of several functional inequalities, including that of quasi-additive functions.

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