Vector Balancing in Lebesgue Spaces

07/10/2020
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by   Victor Reis, et al.
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0
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A tantalizing conjecture in discrete mathematics is the one of Komlรณs, suggesting that for any vectors ๐š_1,โ€ฆ,๐š_n โˆˆ B_2^m there exist signs x_1, โ€ฆ, x_n โˆˆ{ -1,1} so that โˆ‘_i=1^n x_i๐š_i_โˆžโ‰ค O(1). It is a natural extension to ask what โ„“_q-norm bound to expect for ๐š_1,โ€ฆ,๐š_n โˆˆ B_p^m. We prove that, for 2 โ‰ค p โ‰ค q โ‰คโˆž, such vectors admit fractional colorings x_1, โ€ฆ, x_n โˆˆ [-1,1] with a linear number of ยฑ 1 coordinates so that โˆ‘_i=1^n x_i๐š_i_q โ‰ค O(โˆš(min(p,log(2m/n)))) ยท n^1/2-1/p+ 1/q, and that one can obtain a full coloring at the expense of another factor of 1/1/2 - 1/p + 1/q. In particular, for p โˆˆ (2,3] we can indeed find signs ๐ฑโˆˆ{ -1,1}^n with โˆ‘_i=1^n x_i๐š_i_โˆžโ‰ค O(n^1/2-1/pยท1/p-2). Our result generalizes Spencer's theorem, for which p = q = โˆž, and is tight for m = n. Additionally, we prove that for any fixed constant ฮด>0, in a centrally symmetric body K โŠ†โ„^n with measure at least e^-ฮด n one can find such a fractional coloring in polynomial time. Previously this was known only for a small enough constant โ€“ indeed in this regime classical nonconstructive arguments do not apply and partial colorings of the form ๐ฑโˆˆ{ -1,0,1}^n do not necessarily exist.

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