Weak Viterbo conjecture

Let $K\subset\mathbb{R}^{4}\cong\mathbb{H}$ be a convex body invariant under the natural left action of the quaternion group $Q=\{\pm1,\pm i,\pm j,\pm k\}$ . If $\operatorname{sys}(K)$ denotes the minimal action of a closed characteristic on $\partial K$ , then $\operatorname{sys}(K)^{2}\leq 2\operatorname{Vol}(K)$ .

The weak Viterbo conjecture asks for a volume–systolic inequality for convex domains; it was posed by Viterbo in 2000. The unrestricted conjecture is now known to be false, while the symmetry-restricted $Q$ -invariant case tracked here is a narrower question.

Known results

Positive cases include $S^1$ -symmetric convex domains and several other special classes; the general inequality was previously open. (Earlier literature summarized in the cited sources.)
Haim-Kislev and Ostrover constructed a counterexample in every dimension $n\geq 2$ , including a four-dimensional product with systolic ratio $\frac{\sqrt{5}+3}{5}>1$ ; this refutes the unrestricted weak conjecture.

October 2026 claimed $Q$ -symmetric result

Chaidez and Tanny claim the systolic inequality for $Q$ -symmetric convex bodies in dimension four, classify smooth equality cases up to symplectomorphism, and derive results on minimal Reeb-orbit links. The supplied record does not provide independent mathematical assessment, so this remains an unverified claim.

Current status (as of October 2026): The unrestricted weak Viterbo conjecture is false; the four-dimensional $Q$ -invariant case is claimed proved by Chaidez and Tanny but remains unverified.

Weak Viterbo conjecture

Progress summary

Solutions 0