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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},-\...
Iosif Pinelis's user avatar
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 ...
M. Winter's user avatar
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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. ...
Tony Huynh's user avatar
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