Vector Balancing in Lebesgue Spaces
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A tantalizing conjecture in discrete mathematics is the one of Komlós, suggesting that for any vectors 𝐚_1,…,𝐚_n ∈ B_2^m there exist signs x_1, …, x_n ∈{ -1,1} so that ∑_i=1^n x_i𝐚_i_∞≤ O(1). It is a natural extension to ask what ℓ_q-norm bound to expect for 𝐚_1,…,𝐚_n ∈ B_p^m. We prove that, for 2 ≤ p ≤ q ≤∞, such vectors admit fractional colorings x_1, …, x_n ∈ [-1,1] with a linear number of ± 1 coordinates so that ∑_i=1^n x_i𝐚_i_q ≤ O(√(min(p,log(2m/n)))) · 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 ∈ (2,3] we can indeed find signs 𝐱∈{ -1,1}^n with ∑_i=1^n x_i𝐚_i_∞≤ O(n^1/2-1/p·1/p-2). Our result generalizes Spencer's theorem, for which p = q = ∞, and is tight for m = n. Additionally, we prove that for any fixed constant δ>0, in a centrally symmetric body K ⊆ℝ^n with measure at least e^-δ n one can find such a fractional coloring in polynomial time. Previously this was known only for a small enough constant – indeed in this regime classical nonconstructive arguments do not apply and partial colorings of the form 𝐱∈{ -1,0,1}^n do not necessarily exist.
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