On Area-Bisectors of Plane Convex Sets

1963 ◽  
Vol 70 (5) ◽  
pp. 529 ◽  
Author(s):  
Michael Goldberg
Keyword(s):  
Convex Sets ◽  
Plane Convex

Ball characterizations in spaces of constant curvature

2018 ◽  
Vol 55 (4) ◽  
pp. 421-478
Author(s):  
Jesus Jerónimo-Castro ◽  
Endre Makai, Jr.
Keyword(s):  
Convex Sets ◽  
Central Symmetry ◽  
Connected Components ◽  
Plane Convex ◽  
Converse Implication ◽  
Plane Convex Body ◽  
Dual Theorem ◽  
Centrally Symmetric ◽  
Whole Space ◽  
Straight Lines

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C+2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points Sd for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent 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.

Isoperimetric inequalities for the mixed area of plane convex sets

1991 ◽  
Vol 57 (5) ◽  
pp. 501-507 ◽  
Author(s):  
Ulrich Betke ◽  
Wolfgang Weil
Keyword(s):  
Convex Sets ◽  
Isoperimetric Inequalities ◽  
Plane Convex ◽  
Mixed Area

Statistical approximation of plane convex sets

1967 ◽  
Vol 1967 (3-4) ◽  
pp. 113-127 ◽  
Author(s):  
Siv Carlsson ◽  
Ulf Grenander
Keyword(s):  
Convex Sets ◽  
Statistical Approximation ◽  
Plane Convex

scholarly journals Decomposition of plane convex sets. I

1973 ◽  
Vol 47 (2) ◽  
pp. 521-530 ◽  
Author(s):  
Ruth Silverman
Keyword(s):  
Convex Sets ◽  
Plane Convex

scholarly journals Separating plane convex sets.

1990 ◽  
Vol 66 ◽  
pp. 44 ◽  
Author(s):  
Rafael Hope ◽  
Meir Katchalski
Keyword(s):  
Convex Sets ◽  
Plane Convex ◽  
Separating Plane

scholarly journals Almost All String Graphs are Intersection Graphs of Plane Convex Sets

2020 ◽  
Vol 63 (4) ◽  
pp. 888-917
Author(s):  
János Pach ◽  
Bruce Reed ◽  
Yelena Yuditsky
Keyword(s):  
Convex Sets ◽  
Plane Convex ◽  
Intersection Graphs ◽  
String Graphs ◽  
Almost All
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