Envelopes of $\alpha$-sections

TL;DR

This paper systematically studies the envelopes formed by lines that cut a convex body into two parts with a fixed area ratio, exploring their properties and open questions related to fair partitioning.

Contribution

It introduces a detailed analysis of the envelopes of $ extalpha$-sections of convex bodies and investigates their dependence on the ratio $ extalpha$, posing new open questions.

Findings

Envelope properties depend on $ extalpha$

Identifies open questions in fair partitioning

Provides a framework for systematic study

Abstract

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.