An open subset ${\displaystyle S}$  of a Euclidean space ${\displaystyle E}$  is said to satisfy the weak cone condition if, for all ${\displaystyle {\boldsymbol {x}}\in S}$ , the cone ${\displaystyle {\boldsymbol {x}}+V_{{\boldsymbol {e}}({\boldsymbol {x}}),\,h}}$  is contained in ${\displaystyle S}$ . Here ${\displaystyle V_{{\boldsymbol {e}}({\boldsymbol {x}}),h}}$  represents a cone with vertex in the origin, constant opening, axis given by the vector ${\displaystyle {\boldsymbol {e}}({\boldsymbol {x}})}$ , and height ${\displaystyle h\geq 0}$ .

${\displaystyle S}$  satisfies the strong cone condition if there exists an open cover ${\displaystyle \{S_{k}\}}$  of ${\displaystyle {\overline {S}}}$  such that for each ${\displaystyle {\boldsymbol {x}}\in {\overline {S}}\cap S_{k}}$  there exists a cone such that ${\displaystyle {\boldsymbol {x}}+V_{{\boldsymbol {e}}({\boldsymbol {x}}),\,h}\in S}$ .

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