In
mathematics, the cone condition is a property which may be satisfied by a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. Informally, it requires that for each point in the subset a
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines co ...
with vertex in that point must be contained in the subset itself, and so the subset is "non-flat".
Formal definitions
An open subset
of a Euclidean space
is said to satisfy the ''weak cone condition'' if, for all
, the cone
is contained in
. Here
represents a cone with vertex in the origin, constant opening, axis given by the vector
, and height
.
satisfies the ''strong cone condition'' if there exists an
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
of
such that for each
there exists a cone such that
.
References
* {{SpringerEOM , title=Cone condition , id=Cone_condition&oldid=31912 , last=Voitsekhovskii , first=M.I.
Euclidean geometry