Akopyan–Vysotsky perimeter–diameter inequality for planar curves

Let $\gamma$ be a planar curve and let $K$ be a planar convex figure. Suppose that the convex hull of $\gamma$ covers $K$.

Akopyan–Vysotsky conjecture.

$$\operatorname{length}(\gamma) \geq \operatorname{per}(K)-\operatorname{diam}(K).$$

The conjecture concerns a lower bound for the length of a curve whose convex hull contains a prescribed convex figure. The paper's abstract states that this conjecture is proved, so it is included as a solved result.