Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 89-95 (1999)

Correct Soltan-Vasiloi Criterion for Convex Cones

Jaroslav Cel

Warszawska 24c/20, 26-200 Konskie, Poland

Abstract: Let $S$ be a nonempty closed convex locally compact subset of a real locally convex topological linear space $L$ (dim$L \geq 2)$, $S$ different from $L$ and a single point. It is proved among others that if $S$ contains no straight line or $L$ has the complement property, then $S$ is a cone if and only if every three or fewer closed supporting hyperplanes of $S$ have a nonempty intersection. This is a correct version of an absurd theorem of Soltan and Vasiloi established for a convex body in a real quasicompact topological linear space. The possibility of proving such a result under other assumptions on $L$ and $S$ is also discussed.

Classification (MSC91): 52A07, 46A03

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