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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.

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

Bibliographic databases:

Document Type: Article
UDC: 517.98.22
Received: 18.11.2010

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
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\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}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855181482}


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