We construct the first examples of hypersurfaces in any contact manifold of dimension 5 and larger that cannot be C²-approximated by convex hypersurfaces. This contrasts sharply with the foundational result of Giroux in dimension $3$ and the work of Honda-Huang in the C⁰ setting. The main technical step is the construction of a Bonatti-Diaz type blender in the contact setting.
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Julian Chaidez (2024) studied this question.
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