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 [math]\displaystyle{ S }[/math] of a Euclidean space [math]\displaystyle{ E }[/math] is said to satisfy the weak cone condition if, for all [math]\displaystyle{ \boldsymbol{x} \in S }[/math], the cone [math]\displaystyle{ \boldsymbol{x} + V_{\boldsymbol{e}(\boldsymbol{x}),\, h} }[/math] is contained in [math]\displaystyle{ S }[/math]. Here [math]\displaystyle{ V_{\boldsymbol{e}(\boldsymbol{x}),h} }[/math] represents a cone with vertex in the origin, constant opening, axis given by the vector [math]\displaystyle{ \boldsymbol{e}(\boldsymbol{x}) }[/math], and height [math]\displaystyle{ h \ge 0 }[/math].
[math]\displaystyle{ S }[/math] satisfies the strong cone condition if there exists an open cover [math]\displaystyle{ \{ S_k \} }[/math] of [math]\displaystyle{ \overline{S} }[/math] such that for each [math]\displaystyle{ \boldsymbol{x} \in \overline{S} \cap S_k }[/math] there exists a cone such that [math]\displaystyle{ \boldsymbol{x} + V_{\boldsymbol{e}(\boldsymbol{x}),\, h} \in S }[/math].
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
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