LAUSR

Let Ă R4 be a convex domain with smooth boundary .. We use a relation between the extrinsic curvature of . and the Ruelle invariant of the Reeb flow on . to prove that there are constants ą 2 ą 0 independent of . such that 2 ď rup.q ̈ sysp.q1{2 ď Here sysp.q is the systolic ratio of ., i.e. the square of the minimal period of a closed Reeb orbit of . divided by twice the volume of -, and rup.q is the volume-normalized Ruelle invariant. We then construct dynamically convex contact forms on (3 that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomão. These are the first examples of dynamically convex contact 3-spheres that are not strictly contactomorphic to a convex boundary ..