论文标题
大签名子集总和
Large signed subset sums
论文作者
论文摘要
我们研究以下问题：对于给定的$ d \ geq 2 $，$ n \ geq d $和$ k \ leq n $，什么是最大的价值$ c（d，n，k）$，使得从任何一组$ n $ n $ unit vectors中的$ \ m athbb {r}^d $中的任何一组$ n $ unit vectors中，我们可以选择$ k $ k $ k $ vectors $ k $ vectors $ 1 $ 1 $ 1 $ \ \ \ $ c（d，n，k）$？这个问题对于经典矢量总和最小化和平衡问题是双重的，这些问题已经研究了一个多世纪。在一般情况下，我们对$ c（d，n，k）$的渐近估计值均匀。在几种特殊情况下，我们提供了更强的估计：数量$ c（d，n，n）$对应于$ \ ell_p $ - 极化问题，同时确定$ c（d，n，2）$等同于估计矢量系统的连贯性，这是$ p $ frame-frame endergies的特殊情况。当$ n = d+1 $时，为经典的韦尔奇绑定提供了两个新的证明。对于$ n $的巨大值，用于获得$ C（d，n，2）$的罚款估算值。研究了平面案例，给出了$ c（2，n，k）$的急剧界限。最后，我们确定$ c（D，d+1，d+1）$的确切值在某些额外的假设下。
We study the following question: for given $d\geq 2$, $n\geq d$ and $k \leq n$, what is the largest value $c(d,n,k)$ such that from any set of $n$ unit vectors in $\mathbb{R}^d$, we may select $k$ vectors with corresponding signs $\pm 1$ so that their signed sum has norm at least $c(d,n,k)$? The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for $c(d,n,k)$ in the general case. In several special cases, we provide stronger estimates: the quantity $c(d,n,n)$ corresponds to the $\ell_p$-polarization problem, while determining $c(d, n, 2)$ is equivalent to estimating the coherence of a vector system, which is a special case of $p$-frame energies. Two new proofs are presented for the classical Welch bound when $n = d+1$. For large values of $n$, volumetric estimates are applied for obtaining fine estimates on $c(d,n,2)$. Studying the planar case, sharp bounds on $c(2, n, k)$ are given. Finally, we determine the exact value of $c(d,d+1,d+1)$ under some extra assumptions.
