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If we use the Euclidean metric on $\mathbb R^2$, (unit) circles are circles; if we use the Manhattan distance or $\ell^1$ distance, (unit) circles are squares. We can also get polygons with an even number of sides: for instance, circles are hexagons if we define the distance between two points to be the shortest path between them that moves only along the directions of the sides of some fixed triangle.

Is there a metric on $\mathbb R^2$ (that gives the usual topology) for which circles, or at least unit circles, are triangles?

(This question is inspired by though unrelated to this question.)

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    $\begingroup$ Not sure if there is such a metric, but definitely there is no norm whose unit ball is a triangle, because unit balls have to be balanced sets. $\endgroup$
    – GReyes
    Sep 14 at 18:48
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    $\begingroup$ Do you allow nonsymmetric metrics? $\endgroup$
    – Moishe Kohan
    Sep 14 at 18:52
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    $\begingroup$ @MoisheKohan if you allow that the answer is yes (e.g. orient the edges of the triangle from the hexagon example clockwise, and only allow paths that follow edges along their orientation); my question is about "ordinary" metrics. $\endgroup$
    – Mees de Vries
    Sep 14 at 19:25
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    $\begingroup$ @freakish: every unit circle to a triangle? How do you construct such a map? What you outlined is very far from an actual construction. I am not sure it exists. $\endgroup$
    – Moishe Kohan
    Sep 14 at 21:13
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    $\begingroup$ But then this does not answer the posed problem, especially because there is only one unit circle centered at the origin. $\endgroup$
    – Moishe Kohan
    Sep 14 at 21:24

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