  In this note, we prove the following conjecture by A. Akopyan and V.
Vysotsky: If the convex hull of a planar curve $\gamma$ covers a planar convex
figure $K$, then $\operatorname{length}(\gamma) \geq \operatorname{per} (K) -
\operatorname{diam} (K)$. In addition, all cases of equality in this inequality
are studied.
