scholarly journals Boundaries of Smooth Strictly Convex Sets in the Euclidean Plane <i> R</i><sup>2</sup>

2017 ◽  
Vol 07 (02) ◽  
pp. 71-76
Author(s):  
Horst Kramer
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Strictly Convex

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.

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.

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.

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

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:

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.

Sign in / Sign up

Export Citation Format

Share Document