Math 580A: Convex Polytopes
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Fall 2012

I can happily say there are very few errors. Nevertheless:

On page 4, in the second formula for conv(K), "n ≥ 1" has no business being there. It's totally illogical. Question: Where does it belong?

Exercise 0.7.
x(2) should be x(1).
More importantly, the parenthetical remark "(an affine reflection)" is wrong, and it is also irrelevant; you should ignore it.

On page 31, there is a missing end-of-proof symbol at the end of the second paragraph. That is where the proof of Theorem 1.3 ends. The next paragraph is the end of the proof of Theorem 1.2.

The set defined in Definition 1.13 is not topologically closed, therefore cannot be a polyhedron. The mistake is either that the first set definition should read "t ≥ 0", or that the connecting operation should be union instead of (Minkowski) sum.

In the proof of Proposition 2.3(iii), the inequality is not valid "for every λ in R" but only for nonnegative λ, as you can easily see. (That does not affect the proof in an important way.)

On page 67, the end of Sect. 2.5, the statement that simple + simplicial implies simplex is not what Exercise 0.0 says.

On page 68, line 6, homogenization is the opposite of "embedding into"; it is "lifting out of" an affine plane.

The function "sign" that appears in Ch. 6 et seq. is traditionally called the signum function and abbreviated "sgn".

The term "nonnegative dependence" (traditionally, "positive dependence"; see below) ought to be in the index.

The Farkas Lemmas have the following meanings:
1. tells us when an H-polyhedron is empty.
2. tells us when an H-flat does not intersect the closed positive orthant. (By "H-flat" I mean an affine flat that is presented as the intersection of hyperplanes determined by equations)
3. tells us when a half-space contains an H-polyhedron.
4. tells us when a point is not in a V-polyhedron.

Let X = {x¹, x², ..., xⁿ} in R^d. Some novel terminology I recommend to replace the unfortunate choices of "signed vector" and "signed covector":

A weak dependence is a vector λ = (λ₁, ..., λ_n) such that Sum_i λ_i xⁱ = 0 [and Sum_i λ_i = 0, if we mean affine dependence]. The trivial dependence, λ = 0, is a weak dependence. A dependence is a weak dependence that is not trivial. A signed weak dependence is sgn(λ) for a weak dependence λ. This is what the book calls a "signed vector".
A value vector is a vector of values of a linear functional, (f(x¹), ..., f(x^m)). The kernel of f is a hyperplane, which may be the degenerate hyperplane, and need not be a hyperplane generated by points in X. A signed value vector is sgn(f(X)) for a value vector f(X). This is what the book calls a "signed covector".

Here is a table of the terminology and notation I propose (subject to improvement after criticism):

Dep(X)	dependencies (dependency vectors)	circuit dependencies (minimal dependencies: minimal nonempty support)	Min(Dep(X) \ {0})
sgn(Dep(X))	signed dependencies	signed circuits (signs of minimal dependencies)	C(X)
supp(Dep(X))	supports of dependencies	circuits (minimal dependent sets)	C_M(X)

Val(X)	value vectors	cocircuit value vectors (minimal value vectors: minimal nonempty support)	Min(Val(X) \ {0})
sgn(Val(X))	signed value vectors	signed cocircuits (signs of minimal value vectors)	C^*(X)
supp(Val(X))	supports of value vectors	cocircuits (X-hyperplane complements)	C_M^*(X)

The term nonnegative dependence that appears in Sect. 6.2 (p. 156) and elsewhere is traditionally called positive dependence, and that is what I use throughout the course.

Similarly, I claim the terms nonnegative (co)circuit mentioned in Cor. 6.16 would traditionally be called positive (co)circuit if the terms (co)circuit had been known in tradition.
I adopted the term homogeneous (co)circuit for a positive or negative (co)circuit, because they always appear together.