RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Sibirsk. Mat. Zh.: Year: Volume: Issue: Page: Find

 Sibirsk. Mat. Zh., 2011, Volume 52, Number 6, Pages 1346–1356 (Mi smj2279)

On the sectionwise connectedness of a contingent

S. P. Ponomarev, M. Turowska

Institute of Mathematics, Pomeranian Academy in Słupsk, Słupsk, Poland

Abstract: Let $X$ be a real normed space and let $f\colon\mathbb R\to X$ be a continuous mapping. Let $\mathrm T_f(t_0)$ be the contingent of the graph $G(f)$ at a point $(t_0,f(t_0))$ and let $S^+\subset(0,\infty)\times X$ be the “right” unit hemisphere centered at $(0,0_X)$. We show that
1. If $\dim X<\infty$ and the dilation $D(f,t_0)$ of $f$ at $t_0$ is finite then $\mathrm T_f(t_0)\cap S^+$ is compact and connected. The result holds for $\mathrm T_f(t_0)\cap\overline{S^+}$ even with infinite dilation in the case $f\colon[0,\infty)\to X$.
2. If $\dim X=\infty$, then, given any compact set $F\subset S^+$, there exists a Lipschitz mapping $f\colon\mathbb R\to X$ such that $\mathrm T_f(t_0)\cap S^+=F$.
3. But if a closed set $F\subset S^+$ has cardinality greater than that of the continuum then the relation $\mathrm T_f(t_0)\cap S^+=F$ does not hold for any Lipschitz $f\colon\mathbb R\to X$.

Keywords: contingent (tangent cone), dilation, connectedness, compactness, Euclidean space, cardinality.

Full text: PDF file (344 kB)
References: PDF file   HTML file

English version:
Siberian Mathematical Journal, 2011, 52:6, 1069–1078

Bibliographic databases:

Document Type: Article
UDC: 517.98.22

Citation: S. P. Ponomarev, M. Turowska, “On the sectionwise connectedness of a contingent”, Sibirsk. Mat. Zh., 52:6 (2011), 1346–1356; Siberian Math. J., 52:6 (2011), 1069–1078

Citation in format AMSBIB
\Bibitem{PonTur11} \by S.~P.~Ponomarev, M.~Turowska \paper On the sectionwise connectedness of a~contingent \jour Sibirsk. Mat. Zh. \yr 2011 \vol 52 \issue 6 \pages 1346--1356 \mathnet{http://mi.mathnet.ru/smj2279} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2961760} \transl \jour Siberian Math. J. \yr 2011 \vol 52 \issue 6 \pages 1069--1078 \crossref{https://doi.org/10.1134/S0037446611060127} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000298650800012} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855181482}