Cone condition
A condition on a region of Euclidean space expressing some non-flatness property. An open set satisfies the weak cone condition if
for all
, where
is a right circular cone with vertex at the origin of fixed opening
and height
,
, and with axis vector
depending on
. An open set
satisfies the strong cone condition if there exists a covering of the closure
by open sets
such that for any
the cone
is contained in
(the openings of these cones may depend on
). In connection with integral representations of functions and imbedding theorems, anisotropic generalizations of cone conditions have been considered, for example, the weak and strong
-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) |
Cone condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cone_condition&oldid=31912