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 $h\geq 0$ .

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