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As established in a previous post Equilateral Triangles In The Taxicab Space there are exactly $8$ equilateral triangles (of edge length $1$) that can be packed in the $l^1(\mathbb{R}^2)$ unit circle, while there are only $6$ equilateral triangles (of edge length $1$) that can be packed in the $l^2(\mathbb{R}^2)$ unit circle.

My suspicion is that if one were to consider an $l^{1 + \epsilon}(\mathbb{R}^2)$ unit circle, then for sufficiently small epsilon, the number of equilateral triangles which can be packed in the unit circle is still $8$. More so, I suspect that for a specific value of $\epsilon >0$ the $l^{1+ \epsilon}(\mathbb{R}^2)$ unit circle can only contain $7$ equilateral triangles.

Question 1: Is my suspicion correct?

Question 2: Is there a value $\epsilon >0$ such that the $l^{1+ \epsilon}(\mathbb{R}^2)$ unit circle contains exactly $7$ equilateral triangles?

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  • $\begingroup$ Perhaps an easier question to consider might be worth approaching. Notice that in the Taxicab space, $l^1(\mathbb{R}^2)$, one can pack exactly 4 circles of radius $1/2$ inside the circle of radius $1$. For the usual Euclidean space, one can only pack $2$ circles of radius $1/2$ into the the circle of radius $1$. Does there exists an $\epsilon > 0$ so that the $l^{1+\epsilon}(\mathbb{R}^2)$ unit circle contains only $3$ circles or radius $1/2$ ? $\endgroup$ – Drewrl3v Feb 4 '19 at 18:45

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