SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2411-2423, January 2020.
For a collection of $N$ unit vectors ${X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of ${X}$ as the quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, thus giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-\frac p 2} (2-p)^{\frac {p-2} 2}$ which is sharp for $d\leq N\leq 2d$ and $p=1$. We also prove that for $1\leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $p\in[1,p_m]$, and demonstrates an analogous result for all $N$ in the case $d=2$. Finally, we give conjectures on these and other energies.

中文翻译：

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$ p $框架能量的重复最小化

SIAM离散数学杂志，第34卷，第4期，第2411-2423页，2020年1月。
对于$ N $单位向量的集合$ {X} = \ {x_i \} _ {i = 1} ^ N $，定义$ {X} $的$ p $帧能量为$ \ sum_ {i \ neq j} | \ langle x_i，x_j \ rangle | ^ p $。在本文中，我们将最小化此值的问题与另一个优化问题联系在一起，从而为此类能量提供了新的下界。特别是对于$ p <2 $，我们证明该能量至少为$ 2（Nd）p ^ {-\ frac p 2}（2-p）^ {\ frac {p-2} 2} $ $ d \ leq N \ leq 2d $和$ p = 1 $时为锐利。我们还证明，对于$ 1 \ leq m <d $，$ N = d + m $向量的重复正交基构造可将区间p（$ p \ in [1，p_m] $）的能量最小化，并证明了相似的结果在$ d = 2 $的情况下，对于所有$ N $。最后，我们对这些和其他能量进行猜想。