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The answer to both questions is "yes", by the De Bruijn–Erdős theorem.

Not a complete answer, but a sufficient condition. The equation $(e_i - e_j)^TQ(e_i - e_j) = d(i,j)^2$ tells us that $q_{i,i} + q_{j,j} - 2q_{i,j} = d(i,j)^2$, so $q_{i,j} = (q_{i,i} + q_{j,j} - d(i,j …

Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$: $$ f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w …

There are two slightly different questions here (even without discussing John ellipsoids, which I do not know the answer to), so I'll try to be careful in my answer. Question 1: Is the set of absolute …

Iosif Pinelis already gave a nice solution to show that the answer is "no" for sets of infinitely many vectors, and thus for $N$ very large. I'll show that the answer is also "no" even for some set of …

The parallelogram law says that if $\mathcal{V}$ is an inner product space and $\mathbf{v},\mathbf{w} \in \mathcal{V}$, then $$ 2\|\mathbf{v}\|^2 + 2\|\mathbf{w}\|^2 = \|\mathbf{v} + \mathbf{w}\|^2 + …