One property of a planar curve whose convex hull covers a given convex figure

Abstract

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 $\mathrm{length}(\gamma )\ge \mathrm{per}(K)-\mathrm{diam}(K)$. In addition, all cases of equality in this inequality are studied.

arXiv comment10 pages, 4 figures