Английская Википедия:Cone condition

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Шаблон: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 <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

• Шаблон:SpringerEOM