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

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