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Let $N \geq 3$. Does there exist a set in $\mathbb{R}^N$ which is 8-fold symmetric with respect to any coordinate plane $(x_i,x_j)$, and which is not radially symmetric (i.e., not a sphere, ball, spherical shell, or their union, all centred at the origin)?

It seems that in the three-dimensional case, radially symmetric objects are the only examples of 8-fold symmetric sets wrt any coordinate plane. Maybe there is some general way how to establish this result rigorously in all dimensions?

This question is related to and motivated by my previous question in which the 8-fold symmetry assumption wrt some coordinate plane is imposed.

Voliar
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    What does "8-fold symmetric with respect to a 2-plane" mean? – user10354138 Aug 04 '20 at 11:03
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    @user10354138 An object is "8-fold symmetric with respect to a $(x_i.x_j)$-plane" means that the object is invariant under rotation on $\pi/4$-angle between the coordinate axes $x_i$ and $x_j$. – Voliar Aug 04 '20 at 11:56

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No!

Claim: There exists a nonempty countable subset of $\mathbb{R}^N-\{0\}$ which is invariant under all $\pi/4$ rotations on any coordinate 2-planes.

Proof: There are only finitely many coordinate 2-planes, so the group generated by $\pi/4$-rotations on them fixing the other $N-2$ directions is countable. Now just pick any point and look at its orbit. QED.

This clearly is not radially symmetric, since $SO(N)$ is transitive on the sphere $S^{N-1}(r)$ of radius $r$.

As a concrete example, consider $S=(\mathbb{Q}[\sqrt{2}])^N\subset\mathbb{R}^N$. A $\pi/4$-rotation in $(x_{n+1},x_{n+n'+2})$-plane has matrix representation $$ \begin{pmatrix} I_{n}\\ &\frac1{\sqrt2}&&\pm\frac1{\sqrt2}\\ &&I_{n'}\\ &\mp\frac1{\sqrt2}&&\frac1{\sqrt2}\\ &&&&I_{N-n'-n-2} \end{pmatrix} $$ so is an element of $SL_N(\mathbb{Q}[\sqrt2])$ hence must preserve $S$.

user10354138
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  • Thanks! Right, this is certainly a rigorous answer to the question I asked. It is a bit pitty that I formulated the question for general sets, though, since in a problem which I work on I'm interested in solids (or connected open sets) having such symmetry properties. Do you have any clue what happens in such a case? – Voliar Aug 04 '20 at 14:24
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    That's a wonderful question, but it's not the one you asked. You should pose it carefully as a new question. There's no cost to doing so (except that you have to spend the time to formulate it well, but you're the one who SHOULD be spending the time, since you're the one getting the answers!), and having it as a separate question rather than a discussion in the comments makes it much more easily findable for others. – John Hughes Aug 04 '20 at 14:31
  • @JohnHughes Sure, thanks, I'll follow your advice) – Voliar Aug 04 '20 at 19:07