Cone condition

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 $$\boldsymbol{x} + V_{\boldsymbol{e}(\boldsymbol{x}),\, h} \in S$$.