Hi all,
Can anyone point me to some references to the theory of finitely-generated cones in euclidean space? I'd like to know in particular if there is a notion of basis/dimension/linear dependence or so for such cones.
Appreciate any help.
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Sign up to join this communityHi all,
Can anyone point me to some references to the theory of finitely-generated cones in euclidean space? I'd like to know in particular if there is a notion of basis/dimension/linear dependence or so for such cones.
Appreciate any help.
You could take a look at Cones and duality by Aliprantis and Tourky. Specifically Sections 1.6 and 1.7 may have some results that could be of interest to you. Google Books
The authors define a notion of basis of a cone (a set $B$ in the cone so that every vector in the cone is a positive real multiple of an element in $B$; think unit sphere intersected with the cone), but I wonder whether that is what you have in mind. If you mean that every vector in the cone can be written uniquely as a positive linear combination of (extremal) vectors in the cone, then you might want to take a closer look at what are called lattice cones (in finite dimensional spaces, or equivalently, finite dimensional Riesz spaces). Introduction to operator theory in Riesz spaces by Zaanen gives a very gentle introduction to the subject (so don't let the term "operator theory" scare you off). Google Books
People who study toric varieties in algebraic geometry are interested in this kind of notions, since an "affine toric variety" can be completely described by a cone in a euclidean space. Properties of the given cone translate into properties of the variety.
One book about the subject is Fulton, Introduction to toric varieties.
My first recommendation would be Chapter 1 of Fulton's Introduction to Toric Varieties: Google Books
If you need more material I would suggest taking a look at 'Convex cones' by Fuchssteiner and Lusky which is rather good: Google Books
When I was looking for references on related topics a few years ago I found the following book online, which was helpful for picking up terminology, etc. CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY (In particular, chapter 2 covers linear independence and cones at a pretty basic level with plenty of pictures and examples). However, you might wish to look for more standard references on convex geometry and convex analysis. Various books on polytopes also cover related material.
Another classical reference is Oda's Convex Bodies and Algebraic Geometry (no Google Books preview, unfortunately). You might find especially useful the appendix, entitled Geometry of Convex Sets.
If you are interested in the lattice point enumeration aspect, I'd also suggest Computing the Continuous Discretely by Beck and Robins. There's a version of the book available on their website which you can use to preview it.
Grunbaum, convex polytopes: http://books.google.co.uk/books?id=ISHO86XJ1CsC&printsec=frontcover#v=onepage&q&f=false
In some sense this is (part of) the theory of linear programming. If you want a reference for that, check out Bertsimas and Tsitsiklis' Introduction to Linear Optimization.