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Envelopes of $α$-sections

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