In mathematics, the cone condition is a property which may be satisfied by a subset of an Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat".
An open subset
S
of an Euclidean space
E
is said to satisfy the weak cone condition if, for all
x
∈
S
, the cone
x
+
V
e
(
x
)
,
h
is contained in
S
. Here
V
e
(
x
)
,
h
represents a cone with vertex in the origin, constant opening, axis given by the vector
e
(
x
)
, and height
h
≥
0
.
S
satisfies the strong cone condition if there exists an open cover
{
S
k
}
of
S
¯
such that for each
x
∈
S
¯
∩
S
k
there exists a cone such that
x
+
V
e
(
x
)
,
h
∈
S
.
