A reduced planar body with area greater than $\pi {\Delta }^2/4$

Abstract

We construct a reduced planar convex body $R$ with thickness $\Delta (R)=1$ and \[\operatorname{area}(R)=0.786215\ldots>0.785398\ldots=\fracπ{4}.\] Thus $R$ is a counterexample to Lassak's conjectured upper bound $\mathrm{area}\le (\pi /4){\Delta }^2$ 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.

10 pages, 2 figures, plus verification source code