View source for Cone condition

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O.V. Besov,   V.P. Il'in,   S.M. Nikol'skii,   "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Agmon,   "Lectures on elliptic boundary value problems" , v. Nostrand  (1965)</TD></TR></table>