Cone condition

Short description: Condition on subsets of a Euclidean space

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 $\mathit{x}\in S$, the cone $\mathit{x}+V_{\mathit{e}(\mathit{x}),h}$ is contained in $S$. Here $V_{\mathit{e}(\mathit{x}),h}$ represents a cone with vertex in the origin, constant opening, axis given by the vector $\mathit{e}(\mathit{x})$, 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 $\mathit{x}\in \bar{S}\cap S_k$ there exists a cone such that $\mathit{x}+V_{\mathit{e}(\mathit{x}),h}\in S$.

References

• Hazewinkel, Michiel, ed. (2001), "Cone condition", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Cone_condition&oldid=31912