Cone condition

Template:Short description 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 $𝒙\in S$, 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 $h\ge 0$.

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

References

• Template:SpringerEOM