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Given three or more vectors in an inner product space, $x,y,z, \ldots$, I am wondering whether there exist generalisations to the Cauchy-Schwarz inequality: \begin{equation} \left| \langle x,y \rangle \right|^2 \leq \langle x,x \rangle \cdot \langle y,y \rangle. \end{equation}

I understand that one can construct such inequalities following from the positive semi-definiteness of the Gram matrix but I would like to know if there exist other types?

If so, please provide some references. Thanks in advance.

AG1123
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    The Gram matrix generalization is pretty elegant IMO. Is there something specific you don't like about it? –  Nov 18 '15 at 00:51
  • I do not dislike the Gram matrix version. However, I was wondering if there are inequalities that look simpler, even if this is at the expense of being the most general/precise. E.g., for three vectors the inequality following from the Gram matrix is quite long; one (quite trivial) inequality for three would be the one following from combining the Cauch-Schwartz for each pair. Are there any other such inequalities for three vectors or more that are known? – AG1123 Nov 18 '15 at 17:18
  • Moreover, I am wondering whether there are inequalities that one can write for the sum of the squares of norms: $\langle x,x \rangle + \langle y,y \rangle \ldots$ bounded by some combination of all the inner products between them? – AG1123 Nov 18 '15 at 17:23

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