Remark on the Helly number for strongly convex sets on Riemannian manifolds

1981 ◽  
Vol 34 (1) ◽  
pp. 27-29 ◽  
Author(s):  
Norbert Kleinjohann
Keyword(s):  
Riemannian Manifolds ◽  
Convex Sets ◽  
Helly Number ◽  
Strongly Convex

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

scholarly journals A helly number for unions of two boxes inR2

1985 ◽  
Vol 8 (2) ◽  
pp. 267-273 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Convex Sets ◽  
Helly Number ◽  
Boundary Points ◽  
Polygonal Region

LetSbe a polygonal region in the plane with edges parallel to the coordinate axes. If every5or fewer boundary points ofScan be partitioned into setsAandBso thatconv A⋃ conv B⫅S, thenSis a union of two convex sets, each a rectangle. The number5is best possible.Without suitable hypothesis on edges ofS, the theorem fails. Moreover, an example reveals that there is no finite Helly number which characterizes arbitrary unions of two convex sets, even for polygonal regions in the plane.

Maximal $S$-Free Convex Sets and the Helly Number

2016 ◽  
Vol 30 (4) ◽  
pp. 2206-2216 ◽  
Author(s):  
Michele Conforti ◽  
Marco Di Summa
Keyword(s):  
Convex Sets ◽  
Helly Number

scholarly journals Local characterization of strongly convex sets

2013 ◽  
Vol 400 (2) ◽  
pp. 743-750 ◽  
Author(s):  
Alexander Weber ◽  
Gunther Reißig
Keyword(s):  
Convex Sets ◽  
Strongly Convex ◽  
Local Characterization

Convex sets in Riemannian manifolds

1974 ◽  
Vol 14 (5) ◽  
pp. 807-809
Author(s):  
V. A. Sharafutdinov
Keyword(s):  
Riemannian Manifolds ◽  
Convex Sets
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