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@PeterTaylor the consistency of inequations is an assumption, I simply don't consider systems where inequations on the $s_i$'s are inconsistent such as this one. I just want to show that if they are consistent at the level of $s_i$'s, then this can be witnessed by some $x_i$'s.
Here is a proof scheme for existence of an optimal plane: for any orientation, you can compute the optimal plane with this orientation (i.e. you compare all translations of a reference plane), and clearly the optimal among those is reached by at least one plane. Then it amounts to optimizing the function Orientation -> Score. Since Orientations form a compact space, the optimal will be reached again by at least one particular orientation. (and yes I omitted the square root since I just use $d_2$ for minimization, I expected the comment to pop up ^^)
@Gro-Tsen it is mentioned in the blog post above that it can be proven that the f and g summing to id cannot be taken measurable, but he does not provide a proof or reference.
@Gro-Tsen ah sorry I thought it was a classic puzzle, maybe the explanation is missing the fact that the two periodic functions are the projectors on the two subspaces of R. Adding a link to this blog post would provide better context: mathblag.wordpress.com/2013/09/01/sums-of-periodic-functions
Yes I forgot to mention that we want to make this a game of incomplete information in order to make sense of it. So Ann chooses $f$, but Bob does not know what it is, nevertheless he has a probabilistic strategy working with probability 1.
My initial claim was about this "overall" formulation, and I understand now why it is flawed, but I was under the impression that specifying the problem as a game allows to avoid this issue, by explicitely quantifying on all functions in the formulation of the problem and specifying that the probabilities are computed only after this quantification.
Thanks for your answer and for this interesting variant. Unfortunately I must have some kind of block in my thinking, I still don't see why it is not correct to talk about probabilities in a game formulation. In the game you describe, when Bob plays his last move (applying his strategy), he will win with probability 1, and this probability is only computed in the context of the moves already played, ie with a fixed function $f$. The only source of randomness here is Bob's strategy. Maybe the misunderstanding is on whether a stronger claim can be made about an "overall" probability for all f.
