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Suppose I have a finite subset $\mathcal{M}$ of a Banach space $B$ $\mathcal{M}=p_1, \dots, p_n,$ and I create the following ``Gram'' matrix $G_{\mathcal{M}}:$

$$g_{ij} = \frac{\|p_i\|^2 + \|p_j\|^2 - \|p_i - p_j\|^2}{2}.$$ Notice that if $B$ is a Hilbert space, then this is just the usual Gram matrix, and, in particular, positive (semi)-definite. The the question is: what is known about the signature of $G_{\mathcal{M}}?$ Experiment reveals that almost all the eigenvalues are positive when $B$ is $L^1,$ , but it is not obvious to me why that should be so. A secondary (presumably much easier) question is what is the relationship between the signature of this matrix and the Cayley-Menger matrix?

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  • $\begingroup$ You probably know this, but when $B$ is $L^1$ or subspace of $L^1$, it is hypermetric and thus the distance metric $d(x_i,x_j)$ of any $n$ points satisfies a negative-type inequality (Assuad, sur les inégalités valides dans $L^1$). Perhaps these inequalities imply your condition for the "Gram" matrices (?). Nice question. $\endgroup$
    – alvarezpaiva
    Commented Mar 23, 2020 at 14:56
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    $\begingroup$ @alvarezpaiva I did NOT know that. In fact in the (real life) example this came from exactly one eigenvalue is negative, and, bizarrely, the modulus is almost exactly the same as that of the largest positive eigenvalue. $\endgroup$
    – Igor Rivin
    Commented Mar 23, 2020 at 17:51

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