Some changes and suggestions #7
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added 10 commits
February 14, 2014 13:42
The old definition gives a tangent cone that is not necessarily the empty set for a point not in P. This makes a the remark after Brion's theorem incorrect.
The theorem was correct, but it seems a bit strange to require a convex and measurable set.
…ber of iterations of the LLL-Reduce algorithm. I do not believe the old bound to be correct.
…map to the rational functions) that seemed strange.
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This sequence of commits tries to fix a few small mistakes/typos.
I also have some suggestions for further changes:
1.) In Lemma 6.18 which states that we can can cover a convex body K with at most 2^O(d) * N(K,E) translates of an ellipsoid E, it is not mentioned how we can compute an initial point x in the interior of the convex body. It seems that we should use the ellipsoid method, but to do so we might require an initial ball that intersects the interior of K. If something like this is intended the runtime should also depend on the ratio of the volume of the initial ball and its intersection with K.
2.) In general it would be nice to have some more information about how the convex body is given in chapter 6. Presumably it should be given by a separation oracle, and the "runtime-bounds" in many theorems should measure the number of oracle calls and arithmetic operations.
3.) In the chapter 4 it would be nice to know how the set of Lambda-periodic functions that we work with is defined! The way it is now that part is too vague to learn much from it.
4.) In Lemma 7.21 you use the term triangulation of a cone. But you define the term only for polytopes.
5.) After Corollary 4.13 some results about the flatness constant are mentioned. However it is not stated explicitly how the term flatness constant is defined.