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In this paper we are interested in characterizing the standard contact spherein terms of dynamically convex contact manifolds which admit a Liouvillefilling with vanishing symplectic homology. We first observe that if thefilling is flexible, then those contact manifolds are contactomorphic to thestandard contact sphere. We then investigate quantitative geometry of thosecontact manifolds focusing on similarities with the standard contact sphere infiltered symplectic homology.

On dynamically convex contact manifolds and filtered symplectic homology