arXiv:2610.31906v1 Announce Type: new Abstract: We study a construction for producing Finsler metrics on the sphere from convex bodies in $\mathbb{R}^4$ using the quaternions. Among several applications, we prove the weak Viterbo conjecture for convex bodies in $\mathbb{R}^4$ that are invariant under the natural action of the quaternion group $Q$. This is the first instance where Viterbo's systolic inequality is established for convex bodies with a finite group symmetry, addressing a question of Haim-Kislev-Ostrover. Moreover, a smooth $Q$-symmetric convex domain that achieves equality is symplectomorphic to the ball, without any assumption of strict or uniform convexity. Finally, we show that any link of minimal Reeb orbits on the boundary of a uniformly convex $Q$-symmetric convex body is transversely isotopic to a union of Hopf fibers, strengthening a result of Abbondandolo-Edtmair-Kang in the $Q$-symmetric setting.
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