{{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 <math>S</math> of a Euclidean space <math>E</math> is said to satisfy the ''weak cone condition'' if, for all <math>\boldsymbol{x} \in S</math>, the cone <math>\boldsymbol{x} + V_{\boldsymbol{e}(\boldsymbol{x}),\, h}</math> is contained in <math>S</math>. Here <math>V_{\boldsymbol{e}(\boldsymbol{x}),h}</math> represents a cone with vertex in the origin, constant opening, axis given by the vector <math>\boldsymbol{e}(\boldsymbol{x})</math>, and height <math>h \ge 0</math>.
<math>S</math> satisfies the ''strong cone condition'' if there exists an open cover <math>\{ S_k \}</math> of <math>\overline{S}</math> such that for each <math>\boldsymbol{x} \in \overline{S} \cap S_k</math> there exists a cone such that <math>\boldsymbol{x} + V_{\boldsymbol{e}(\boldsymbol{x}),\, h} \in S</math>.
== References == * {{SpringerEOM | title=Cone condition | id=Cone_condition&oldid=31912 | last=Voitsekhovskii | first=M.I.}}
Category:Euclidean geometry