A reduced planar body with area greater than $πΔ^2/4$

2026-06-26

AI Summary

This study addresses Lassak’s long-standing conjecture on planar reduced bodies, which posits an upper bound of $\operatorname{area} \leq (\pi/4)\Delta^2$ for the area in terms of thickness $\Delta$. The authors refute this conjecture by explicitly constructing a convex reduced body of thickness $\Delta = 1$ whose area exceeds $\pi/4$. Employing the support function method together with elementary geometric analysis of width, area, and contact points, they present the first counterexample with area $0.786215\ldots$, strictly greater than $\pi/4 \approx 0.785398$. This construction disproves the conjectured upper bound and provides a more precise understanding of the maximal possible area of planar reduced bodies.

Abstract

We construct a reduced planar convex body $R$ with thickness $Δ(R)=1$ and \[\operatorname{area}(R)=0.786215\ldots>0.785398\ldots=\fracπ{4}.\] Thus $R$ is a counterexample to Lassak's conjectured upper bound $\operatorname{area}\le(π/4)Δ^2$ for planar reduced bodies. The construction is given by an explicit support function, and the proofs use only elementary support-function, width, area, and contact-point computations.

Problem

• reduced body
• planar convex body
• area upper bound
• thickness
• Lassak's conjecture

Innovation

• reduced convex body
• area upper bound
• thickness
• support function
• counterexample