M. Studeny: Convex cones in finite-dimensional real vector spaces. Kybernetika 29 (1993), n. 2, pp. 180-200.

Equivalent characterizations of pointed cones, pyramids and rational pyramids in a finite-dimensional real vector space are given. Special class of regular cones, corresponding to ``continuous linear" quasiorderings of integer vectors is introduced and equivalently characterized. Two different ways of determining of vector quasiorderings are studied: establishing (i.e. prescribing a set of `positive' vectors) and inducing through scalar product. The existence of the least finite set of normalized integer vectors establishing every finitely establishable (or equivalently finitely inducable) ordering of integer vectors is shown. For every quasiordering of integer vectors established by a finite exhaustive set there exists the least finite set of normalized integer vectors inducing it and the elements of this set can be distinguished by corresponding `positive' integer vectors.

The results of the paper are necessary for technical proofs in the series of papers :
M. Studeny: Description of structures of stochastic conditional independence by means of faces and imsets. International Journal of General Systems 23 (1994/5), n. 2-4, p. 123-137, pp. 201-219, pp. 323-341.

AMS classification 52A20, 52B11

closed convex cone
dual cone
pointed cone
regular cone
extreme ray
(rational) pyramid

A pdf copy (converted postscript version) (272kB) is available.