In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the cone condition is a property which may be satisfied by a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. 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 con ...
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_\alpha\s ...
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