Envelopes of $$\alpha $$ α -Sections

Abstract

Let K be a planar convex body af area |K|, and take 0 < α < 1. An α-section of K is a line cutting K into two parts, one of which has area α|K|. This article presents a systematic study of the envelope of α-sections and its dependence on α. Several open questions are asked, one of them in relation to a problem of fair partitioning.

Fair Partitioning by Straight Lines

“…Our next result, Theorem 4, concerns a fair pizza partition problem using the cutting rule. It has been already mentioned in [6] as a consequence of Theorem 1, but without a proof of implication. Here we give a proof, and thus confirm the result.…”

“…The area of A ∈ K is denoted by |A| and its boundary is denoted by ∂A. Following [6], what we call a pizza is a pair (A, B) of two nested planar convex bodies A ⊆ B ⊂ R 2 . We call A the topping and B the dough.…”