## New answers tagged counterexamples

23
votes

Accepted

### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

A counterexample is given by the following five points:
$$(0,0),(1,0),
\Big(-\frac{64867}{77629},\frac{3389}{60094}\Big),
\Big(\frac{5981}{56176},\frac{32211}{34172}\Big),
\Big(\frac{5925}{117812},-\...

1
vote

### Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$

For an isometric embedding the distances in the graph must satisfy at least the triangle inequality. Let me assume that this is implicit in your notion of "metric graph". But then with more ...

8
votes

Accepted

### Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$

With the $\ell_\infty$-norm this is true. For example, it is a classic theorem of Fréchet that every $n$-point metric space embeds in $\ell_\infty^{n-1}$. The required embedding $f$ is easy to define. ...

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