The Viterbo conjecture is the proposed volume-capacity inequality
 |
for every convex body 
and every normalized symplectic
capacity 
(Viterbo 2000). Here 
denotes Euclidean space volume,
and 
is a factorial. Equality holds for a ball.
A stronger statement asserts that all normalized symplectic
capacities coincide on convex bodies.
Haim-Kislev and Ostrover (2026) refuted the inequality for every 
using the Ekeland-Hofer-Zehnder symplectic
capacity 
.
Their four-dimensional counterexample is a Cartesian product of two suitably positioned regular
pentagons. It also refutes the stronger capacity-coincidence
statement.
A sharp positive result survives under a specific finite symmetry. Chaidez and Tanny (2026) proved that every convex body 
, where 
is the quaternion algebra,
invariant under right multiplication by the quaternion
group satisfies
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For smooth such convex bodies, equality holds iff their interiors are related to a ball by a symplectic diffeomorphism. Chaidez and Tanny (2026) report that ChatGPT assisted with the quaternion construction and proposed a chord bound used in the argument.
