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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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Since you're asking for a list of references, you might want to make the question community wiki –  j.c. Mar 23 '10 at 18:54
    
I guess the 1-skeleton of the cone is the closest thing to a 'basis' for cones: picking a representative vector for each 1-simplex, any vector in the cone can be writen (non-uniquely) as a positive linear combination of these vectors. –  J.C. Ottem Apr 19 '10 at 16:19
    
If you care about the integer points in the cone, there is the notion of Hilbert basis: en.wikipedia.org/wiki/Hilbert_basis_(linear_programming) which might be what you're looking for. –  Steven Sam Apr 19 '10 at 21:04

8 Answers 8

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.

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Fulton defines the dimension of the convex polyhedral cone $C=[\sum_{i=1}^m\alpha_i v_i| \alpha_i\geq0]\subset\mathbb{R}^n$ as the dimension of the linear space $C+(-C)$. He does not define any notion of "basis" for the cone $C$. Does anybody know if there is some notion of basis for a cone? –  Shake Baby Mar 23 '10 at 21:22
    
There is a concept of a generating set for a cone (take positive linear combinations of the vectors in that set to generate the cone) and accordingly, a concept of conic independence. See 2.5 in the book in my answer for one fairly detailed source. –  j.c. Mar 25 '10 at 15:05

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

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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

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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.

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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.

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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.

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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.

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