On Archimedean ordered vector spaces and a characterization of simplices

Gerhard Gierz, Boris Shekhtman
1992 Proceedings of the American Mathematical Society  
We show that a convex subset K of a linear space is a simplex if and only if it is line compact and every nonempty intersection of two translates of K is a homothet of K . This answers a problem posed by Rosenthal. The proof uses a reformulation of this problem in terms of Archimedean ordered spaces Received by the editors December 4, 1990 and, in revised form, February 21, 1991. 1991 Mathematics Subject Classification. Primary 46A55, 46A40. Key words and phrases. Simplices, Archimedean ordered
more » ... spaces, sublattices of C(K). 1 K is line-compact if the intersection of every line with K is compact. 2A homothet of K is a set of the form a + r • K , where r > 0 is a positive constant.
doi:10.1090/s0002-9939-1992-1095222-9 fatcat:bfavnwydu5haxlo6onx7drhi34