Suppose that $a,b,c$ are orthonormal vectors in $\mathbb{R}^{n}$ ($n\ge 3$). How large can $|\sum_{k=1}^{n}a_{k}b_{k}c_{k}|$ be as a function of $n$?
I'm also generally interested in any inequalities or just theory/background information about sums of this form.
The Holder inequality gives that
$$|\sum_{k=1}^{n}a_{k}b_{k}c_{k}|\le \|{a}\|_{3}\|{b}\|_{3}\|{c}\|_{3}$$
for any vectors $a,b,c\in \mathbb{R}^{n}$ or $\mathbb{C}^{n}$. The question was whether orthogonality of $a,b,c$ might allow for a smaller lower bound. It does not.
For an example in $\mathbb{R}^{4}$, take $a = (1,-1,-1,1), b=(-1,1,-1,1), c=(-1,-1,1,1)$. Then $\sum_{k=1}^{n}a_{k}b_{k}c_{k} = 4 = \|a\|_{3}\|b\|_{3}\|c\|_{3}$
In $\mathbb{C}^{n}$, certain Fourier basis vectors give an example where the worst case is achieved. Let $3|n$, and let $k_{1} = n/3$, $k_{2} = 2n/3$. Let $\omega$ be a primitive $n$th root of unity. Then take $a = (\omega^{k_{1}},\omega^{2k_{1}},\ldots, \omega^{nk_{1}}), b =(\omega^{k_{2}},\omega^{2k_{2}},\ldots, \omega^{nk_{2}}), c = (1,\ldots, 1)$. Then
$$ \sum_{k=1}^{n}a_{k}b_{k}c_{k} = \sum_{k=1}^{n}1 = n = \|a\|_{3}\|b\|_{3}\|c\|_{3} $$ There are simpler examples in $\mathbb{C}^{n}$, but this generalizes easily.