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
postscript version (294kB) is available.