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 .
