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Cone condition

From Wikipedia, the free encyclopedia

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

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An open subset of a Euclidean space is said to satisfy the weak cone condition if, for all , the cone is contained in . Here represents a cone with vertex in the origin, constant opening, axis given by the vector , and height .

satisfies the strong cone condition if there exists an open cover of such that for each there exists a cone such that .

References

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  • Voitsekhovskii, M.I. (2001) [1994], "Cone condition", Encyclopedia of Mathematics, EMS Press