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Difference between revisions of "Cone condition"

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A condition on a region of Euclidean space expressing some non-flatness property. An open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246001.png" /> satisfies the weak cone condition if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246002.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246003.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246004.png" /> is a right circular cone with vertex at the origin of fixed opening <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246005.png" /> and height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246007.png" />, and with axis vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246008.png" /> depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c0246009.png" />. An open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460010.png" /> satisfies the strong cone condition if there exists a covering of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460011.png" /> by open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460012.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460013.png" /> the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460014.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460015.png" /> (the openings of these cones may depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460016.png" />). In connection with integral representations of functions and imbedding theorems, anisotropic generalizations of cone conditions have been considered, for example, the weak and strong <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024600/c02460018.png" />-horn conditions (see [[#References|[1]]]), the cube condition, etc.
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A condition on a region of Euclidean space expressing some non-flatness property. An open set $G\subset E^n$ satisfies the weak cone condition if $x+V(e(x),H)\subset G$ for all $x\in G$, where $V(e(x),H)$ is a right circular cone with vertex at the origin of fixed opening $\epsilon$ and height $H$, $0\leq H\leq\infty$, and with axis vector $e(x)$ depending on $x$. An open set $G$ satisfies the strong cone condition if there exists a covering of the closure $\bar G$ by open sets $G_k$ such that for any $x\in\bar G\cap G_k$ the cone $x+(V(e(x),H)$ is contained in $G$ (the openings of these cones may depend on $k$). In connection with integral representations of functions and imbedding theorems, anisotropic generalizations of cone conditions have been considered, for example, the weak and strong $l$-horn conditions (see [[#References|[1]]]), the cube condition, etc.
  
 
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Latest revision as of 15:39, 24 April 2014

A condition on a region of Euclidean space expressing some non-flatness property. An open set $G\subset E^n$ satisfies the weak cone condition if $x+V(e(x),H)\subset G$ for all $x\in G$, where $V(e(x),H)$ is a right circular cone with vertex at the origin of fixed opening $\epsilon$ and height $H$, $0\leq H\leq\infty$, and with axis vector $e(x)$ depending on $x$. An open set $G$ satisfies the strong cone condition if there exists a covering of the closure $\bar G$ by open sets $G_k$ such that for any $x\in\bar G\cap G_k$ the cone $x+(V(e(x),H)$ is contained in $G$ (the openings of these cones may depend on $k$). In connection with integral representations of functions and imbedding theorems, anisotropic generalizations of cone conditions have been considered, for example, the weak and strong $l$-horn conditions (see [1]), the cube condition, etc.

References

[1] O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)


Comments

References

[a1] S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965)
How to Cite This Entry:
Cone condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cone_condition&oldid=31912
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article