On reduced polytopes

A convex body $R$ in ${\mathbb{R}}^d$ is called reduced if the minimal width $\mathrm{\Delta }(R^{\mathrm{\prime }})$ of each convex body $R^{\mathrm{\prime }}\subset R$ different from $R$ is strictly smaller than the minimal width $\mathrm{\Delta }(R)$ of $R$. In this article we construct a reduced polytope in ${\mathbb{R}}^3$, i.e. we answer the following question posed by Lassak: do there exist reduced polytopes in ${\mathbb{R}}^d$, $d⩾3$? Also, we prove some properties of reduced polytopes in $\mathbb R^3$.