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 ${\displaystyle S}$ of a Euclidean space ${\displaystyle E}$ is said to satisfy the weak cone condition if, for all ${\displaystyle {\boldsymbol {x}}\in S}$, the cone ${\displaystyle {\boldsymbol {x}}+V_{{\boldsymbol {e}}({\boldsymbol {x}}),\,h}}$ is contained in ${\displaystyle S}$. Here ${\displaystyle V_{{\boldsymbol {e}}({\boldsymbol {x}}),h}}$ represents a cone with vertex in the origin, constant opening, axis given by the vector ${\displaystyle {\boldsymbol {e}}({\boldsymbol {x}})}$, and height ${\displaystyle h\geq 0}$.

${\displaystyle S}$ satisfies the strong cone condition if there exists an open cover ${\displaystyle \{S_{k}\}}$ of ${\displaystyle {\overline {S}}}$ such that for each ${\displaystyle {\boldsymbol {x}}\in {\overline {S}}\cap S_{k}}$ there exists a cone such that ${\displaystyle {\boldsymbol {x}}+V_{{\boldsymbol {e}}({\boldsymbol {x}}),\,h}\in S}$.

References

• Voitsekhovskii, M.I. (2001) [1994], "Cone condition", Encyclopedia of Mathematics, EMS Press