Cone condition explained

In mathematics, the cone condition is a property which may be satisfied by a subset of a 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".

Formal definitions

An open subset

S of a Euclidean space E is said to satisfy the weak cone condition if, for all \boldsymbol{x} \in S , the cone \boldsymbol{x} + V _{\boldsymbol{e} (\boldsymbol{x}), h} is contained in S . Here V _{\boldsymbol{e} (\boldsymbol{x}),h} represents a cone with vertex in the origin, constant opening, axis given by the vector \boldsymbol{e}(\boldsymbol{x}) , and height h \ge 0 . S satisfies the strong cone condition if there exists an open cover \{ S _{k} \} of \overline{S} such that for each \boldsymbol{x} \in \overline{S} \cap S _{k} there exists a cone such that \boldsymbol{x} + V _{\boldsymbol{e} (\boldsymbol{x}), h} \in S