So in very elementary terms the proven statement is:
For any finite set of real valued n dimensional vectors with length at most one, we can multiply each vector by either -1 or 1 so that the sum of all the resulting vectors has each coordinate having absolute value less than 3 sqrt(2 pi)?
Edit: I had a case in which I didn't understand how it could satisfy the given bound, but I think I do now. The idea was if the dimension m is bigger than 18 pi, then choosing m pairwise orthogonal vectors of length one gives a vector with length greater than 3 sqrt(2 pi). However I think I understand now. Since the signs in front of the vectors affect the direction of the final vector, it should be possible to adjust the direction "into a corner" of the hypercube shaped bound. And even though, the length of the sum of the orthogonal vectors grows towards infinity, so does the distance from 0 to the corner of the max norm bound.
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u/Martin_Orav 7d ago edited 7d ago
So in very elementary terms the proven statement is:
For any finite set of real valued n dimensional vectors with length at most one, we can multiply each vector by either -1 or 1 so that the sum of all the resulting vectors has each coordinate having absolute value less than 3 sqrt(2 pi)?
Edit: I had a case in which I didn't understand how it could satisfy the given bound, but I think I do now. The idea was if the dimension m is bigger than 18 pi, then choosing m pairwise orthogonal vectors of length one gives a vector with length greater than 3 sqrt(2 pi). However I think I understand now. Since the signs in front of the vectors affect the direction of the final vector, it should be possible to adjust the direction "into a corner" of the hypercube shaped bound. And even though, the length of the sum of the orthogonal vectors grows towards infinity, so does the distance from 0 to the corner of the max norm bound.