(n,m)-fold covers of spheres
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Tartalom: http://dx.doi.org/10.1134/S0081543815010150
Archívum: MTA Könyvtár
Gyűjtemény: Status = Published

Type = Article
Cím:
(n,m)-fold covers of spheres
Létrehozó:
Bárány, Imre
Fabila-Monroy, R.
Vogtenhuber, B.
Dátum:
2015
Téma:
QA Mathematics / matematika
Tartalmi leírás:
A well-known consequence of the Borsuk-Ulam theorem is that if the d-dimensional sphere Sd is covered with less than d + 2 open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the d-dimensional sphere n times, with the additional property that the northern hemisphere is covered m > n times. We prove that if the open northern hemisphere is to be covered m times, then at least ?(d ? 1)/2? + n + m and at most d + n + m sets are needed. For the case of n = 1 and d ? 2, this number is equal to d + 2 if m ? ?d/2? + 1 and equal to ?(d ? 1)/2? + 2 + m if m > ?d/2? + 1. If the closed northern hemisphere is to be covered m times, then d + 2m ? 1 sets are needed; this number is also sufficient. We also present results on a related problem of independent interest. We prove that if Sd is covered n times with open sets not containing a pair of antipodal points, then there exists a point that is covered at least ?d/2? + n times. Furthermore, we show that there are covers in which no point is covered more than n + d times. © 2015, Pleiades Publishing, Ltd.
Típus:
Article
PeerReviewed
Formátum:
text
Azonosító:
Bárány, Imre and Fabila-Monroy, R. and Vogtenhuber, B. (2015) (n,m)-fold covers of spheres. Proceedings of the Steklov Institute of Mathematics, 288 (1). pp. 203-208. ISSN 0081-5438
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