A Condition for Sets in R 3 to be Cones

Let S be a nonempty proper subset of R ³ . A point y in cl, S is clearly R -visible from a point x via S if and only if there exists a neighborhood N of y such that S contains all closed half-lines emanating from x through points of . The following Krasnosel'skii-type theorem is proved—if every two boundary points of S are clearly R -visible via S from a common point, then S is a cone. This improves an earlier result with the number three and answers an open question. Received March 18, 1998, and in revised form June 2, 1999.