RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Zametki, 2015, Volume 98, Issue 5, Pages 643–650 (Mi mz10612)  

Quantitative Expressions for the Connectedness of Sets in ${\mathbb R}^n$

P. A. Borodin, O. N. Kosukhin

Lomonosov Moscow State University

Abstract: We prove that, for two arbitrary points $a$ and $b$ of a connected set $E\subset\nobreak {\mathbb R}^n$ ($n\ge 2$) and for any $\varepsilon>0$, there exist points $x_0=a$, $x_2,…,x_p=b$ in $E$ such that
$$ \|x_1-x_0\|^n+…+\|x_p-x_{p-1}\|^n<\varepsilon. $$
We prove that the exponent $n$ in this assertion is sharp. The nonexistence of a chain of points in $E$ with
$$ \|x_1-x_0\|^\alpha+…+\|x_p-x_{p-1}\|^\alpha<\varepsilon $$
for some $\alpha\in (1,n)$ proves to be equivalent to the existence of a nonconstant function $f\colon E\to {\mathbb R}$ in the class $\operatorname{Lip}_\alpha(E)$. For each such $\alpha$, we construct a curve $E(\alpha)$ of Hausdorff dimension $\alpha$ in ${\mathbb R}^n$ and a nonconstant function $f\colon E(\alpha)\to {\mathbb R}$ such that $f\in\operatorname{Lip}_\alpha(E(\alpha))$.

Keywords: connectedness, Hausdorff dimension, Lipschitz property, Euclidean space.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00510
14-01-91158
15-01-08335
15-01-08335
Ministry of Education and Science of the Russian Federation НШ-3682.2014.1
Dynasty Foundation

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm10612

Full text: PDF file (433 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2015, 98:5, 707–713

Bibliographic databases:

UDC: 515.125+517.518.26
Received: 30.10.2014
Revised: 25.03.2015

Citation: P. A. Borodin, O. N. Kosukhin, “Quantitative Expressions for the Connectedness of Sets in ${\mathbb R}^n$”, Mat. Zametki, 98:5 (2015), 643–650; Math. Notes, 98:5 (2015), 707–713

Citation in format AMSBIB
\Bibitem{BorKos15}
\by P.~A.~Borodin, O.~N.~Kosukhin
\paper Quantitative Expressions for the Connectedness of Sets in~${\mathbb R}^n$
\jour Mat. Zametki
\yr 2015
\vol 98
\issue 5
\pages 643--650
\mathnet{http://mi.mathnet.ru/mz10612}
\crossref{https://doi.org/10.4213/mzm10612}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3438521}
\elib{http://elibrary.ru/item.asp?id=24850189}
\transl
\jour Math. Notes
\yr 2015
\vol 98
\issue 5
\pages 707--713
\crossref{https://doi.org/10.1134/S0001434615110012}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000369701000001}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84953236028}


Linking options:
  • http://mi.mathnet.ru/eng/mz10612
  • https://doi.org/10.4213/mzm10612
  • http://mi.mathnet.ru/eng/mz/v98/i5/p643

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математические заметки Mathematical Notes
    Number of views:
    This page:338
    Full text:49
    References:34
    First page:37

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019