Cone condition

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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 [1]), the cube condition, etc.


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



[a1] S. Agmon, "Lectures on elliptic boundary value problems" , v. Nostrand (1965)
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Cone condition. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article