Unions of Two Convex Sets

1963 ◽  
Vol 15 ◽  
pp. 152-156 ◽  
Author(s):  
William L. Stamey ◽  
J. M. Marr
Keyword(s):  
Closed Subset ◽  
Euclidean Plane ◽  
Convex Sets ◽  
Point Property ◽  
Isolated Points

Valentine (1, Theorems 2 and 3) has defined a three-point property which he called P3 and has shown that a closed subset of the euclidean plane possessing this property is expressible as the union of at most three convex sets. He also showed that if the number of isolated points of local non-convexity of such a set is one, finite and even, or infinite, the set is the union of two convex sets. In this paper we give properties which, together with Valentine's results, characterize those subsets of a plane which may be represented as a union of two closed, convex sets.

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals On Fixed Point Property under Lipschitz and Uniform Embeddings

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Jichao Zhang ◽  
Lingxin Bao ◽  
Lili Su
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Fixed Point Property ◽  
Point Property ◽  
Lipschitz Mappings ◽  
Affine Mappings ◽  
Reflexive Space ◽  
Gateaux Differentiability

We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.

A Decomposition for Sets Having a Segment Convexity Property

1980 ◽  
Vol 32 (1) ◽  
pp. 21-26
Author(s):  
Marilyn Breen
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Convex Set ◽  
Finite Union ◽  
Combinatorial Property ◽  
Convexity Property ◽  
Convexity Condition ◽  
Line Segments ◽  
Isolated Points ◽  
Decomposition Theorems

Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments.

Touching Convex Sets in the Plane

1994 ◽  
Vol 37 (4) ◽  
pp. 495-504 ◽  
Author(s):  
Meir Katchalski ◽  
János Pach
Keyword(s):  
Positive Constant ◽  
Euclidean Plane ◽  
Convex Sets ◽  
Nonempty Intersection ◽  
The Other ◽  
Plane Convex ◽  
Straight Line

AbstractTwo subsets of the Euclidean plane touch each other if they have a point in common and there is a straight line separating one from the other.It is shown that there exists a positive constant c such that if are families of plane convex sets with for some k ≥ 1 and if every touches every then either contains k members having nonempty intersection.

A condition for a compact plane set to be a union of finitely many convex sets

1974 ◽  
Vol 76 (1) ◽  
pp. 61-66 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
The Real ◽  
Compact Plane ◽  
Convex Subsets

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

scholarly journals Two inequalities for convex sets in the plane

1978 ◽  
Vol 19 (1) ◽  
pp. 131-133 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Convex Set ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded, closed, convex set in the euclidean plane having diameter d, width w, inradius r, and circumradius R. We show thatandwhere both these inequalities are best possible.

scholarly journals Fixed point properties for semigroups of nonexpansive mappings on convex sets in dual Banach spaces

2018 ◽  
Vol 17 (1) ◽  
pp. 67-87
Author(s):  
Anthony To-Ming Lau ◽  
Yong Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Fixed Point Properties ◽  
Dual Banach Space ◽  
Standing Problem

Abstract It has been a long-standing problem posed by the first author in a conference in Marseille in 1990 to characterize semitopological semigroups which have common fixed point property when acting on a nonempty weak* compact convex subset of a dual Banach space as weak* continuous and norm nonexpansive mappings. Our investigation in the paper centers around this problem. Our main results rely on the well-known Ky Fan’s inequality for convex functions.

On the Semigroup of Probability Measures of a Locally Compact Semigroup

1987 ◽  
Vol 30 (2) ◽  
pp. 142-146 ◽  
Author(s):  
James C.S. Wong
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Compact Semigroup ◽  
Continuous Functions ◽  
Convolution Semigroup ◽  
Probability Measures ◽  
Point Property ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Uniformly Continuous

AbstractWe show that a locally compact semigroup S is topological left amenable iff a certain space of left uniformly continuous functions on the convolution semigroup of probability measures M0(S) on S is left amenable or equivalently iff the convolution semigroup M0(S) has the fixed point property for uniformly continuous affine actions on compact convex sets.

scholarly journals A family of inequalities for convex sets

1979 ◽  
Vol 20 (2) ◽  
pp. 237-245 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Convex Set ◽  
Upper Bounds ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r, and R respectively. We establish a number of best possible upper bounds for (w−2r)d, (w−2r)R,(w−2r)p, (w−2r)A in terms of w and r. Examples are:

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