Vector balancing in Lebesgue spaces
Vector balancing in Lebesgue spaces
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勒贝格空间中的向量平衡
DOI：
10.1002/rsa.21113
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发表时间：
2020
影响因子：
1
通讯作者：
T. Rothvoss
中科院分区：
文献类型：
--
作者：
Victor Reis;T. Rothvoss
Komlós的猜想表明，对于任何向量A1，…，An∈B2M$$ {\ BoldSymbol {a}} _ 1，\ dots，{\ boldsymbol {a}}} _ n \ in {b} ，…，xn∈{−1,1} $$ {x} _1，\ dots，{x} _n \ in \ left \ left \ { - 1,1 \ right \} $ $ ≤o（1）$$ {\ left \ vert {\ sum} _ {i = 1}^n {x} _i {\ boldsymbol {a}} _ i \ right \ right \ vert} _ {\ right \ vert} _ {\ infty} 1） $$。 Bowsymbol {a}} _ n \ in {b} _p^m $$。我们通过证明任何δ> 0 $$ \ delta> 0来实现这一目标$$，对称的凸形主体k⊆ℝn$$ k \ subseteq {\ mathbb {r}}}}^n $$，带有高斯测量至少E-Δn$$ {e}^{ - \ delta n} $$部分着色。
The Komlós conjecture suggests that for any vectors a1,…,an∈B2m$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_2^m $$ there exist x1,…,xn∈{−1,1}$$ {x}_1,\dots, {x}_n\in \left\{-1,1\right\} $$ so that ‖∑i=1nxiai‖∞≤O(1)$$ {\left\Vert {\sum}_{i=1}^n{x}_i{\boldsymbol{a}}_i\right\Vert}_{\infty}\le O(1) $$ . It is a natural extension to ask what ℓq$$ {\ell}_q $$ ‐norm bound to expect for a1,…,an∈Bpm$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_p^m $$ . We prove a tight partial coloring result for such vectors, implying a nearly tight full coloring bound. As a corollary, this implies a special case of Beck–Fiala's conjecture. We achieve this by showing that, for any δ>0$$ \delta >0 $$ , a symmetric convex body K⊆ℝn$$ K\subseteq {\mathbb{R}}^n $$ with Gaussian measure at least e−δn$$ {e}^{-\delta n} $$ admits a partial coloring. Previously this was known only for a small enough δ$$ \delta $$ . Additionally, we show that a hereditary volume bound suffices to provide such Gaussian measure lower bounds.