Envelopes of $\alpha$-sections

Let $K$ be a planar convex body af area $|K|$, and take $0 \textless{} \alpha \textless{} 1$.An $\alpha$-section of $K$ is a line cutting $K$ into two parts, one of whichhas area $\alpha|K|$. This article presents a systematic study of the envelope of $\alpha$-sectionsand its dependence on $\alpha$. Several open questions are asked, one of themin relation to a problem of fair partitioning.